Online Appendix Aggregate Implications of Innovation Policy

Andrew Atkeson* Ariel Burstein ^(†){ }^{\dagger}

June 2018

Contents (Appendix)

C Quantitative analysis 2
C. 1 Impact elasticities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
C. 2 Data and calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
C. 3 Calibration of additional parameters to solve nonlinear transition dynamics 7
C. 4 Solving the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
C. 5 Additional numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
D Discussion of models not nested in our framework 26
E Variations of baseline model 28
E. 1 Occupation choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
E. 2 Goods and labor used as inputs in research . . . . . . . . . . . . . . . . . . . 29
E. 3 Klette and Kortum (2004) specification . . . . . . . . . . . . . . . . . . . . . . 31
E. 4 Alternative specification of incumbent technology to acquire new products 33

C Quantitative analysis

Section C. 1 provides alternative expressions to measure impact elasticities. Section C. 2 provides additional details of the calibration procedure for our analytic elasticities. Section C. 3 describes how we calibrate additional parameters that are required to solve the model nonlinearly. Section C. 4 provides the equations that are used to solve the model (BGP and transition dynamics). Section C. 5 provides additional details for the results of Section 7 .

C. 1 Additional formulas for impact elasticities

By equation (68),
(85) Θ e = 1 ρ 1 ( η e δ e ζ ( x ¯ c ) ) x ¯ e exp ( g ¯ Z ) ρ 1 Y ¯ r x ¯ e (85) Θ e = 1 ρ 1 η e δ e ζ x ¯ c x ¯ e exp g ¯ Z ρ 1 Y ¯ r x ¯ e {:(85)Theta_(e)=(1)/(rho-1)((eta_(e)-delta_(e)zeta( bar(x)_(c))) bar(x)_(e))/(exp ( bar(g)_(Z))^(rho-1))( bar(Y)_(r))/( bar(x)_(e)):}\begin{equation*} \Theta_{e}=\frac{1}{\rho-1} \frac{\left(\eta_{e}-\delta_{e} \zeta\left(\bar{x}_{c}\right)\right) \bar{x}_{e}}{\exp \left(\bar{g}_{Z}\right)^{\rho-1}} \frac{\bar{Y}_{r}}{\bar{x}_{e}} \tag{85} \end{equation*}(85)Θe=1ρ1(ηeδeζ(x¯c))x¯eexp(g¯Z)ρ1Y¯rx¯e
Given the definition of G G GGG in (28), we have that
exp ( G ( x ¯ c , x ¯ m , x ¯ e ) ) ρ 1 exp ( G ( x ¯ c , x ¯ m , 0 ) ) ρ 1 = ( η e δ e ζ ( x ¯ c ) ) x ¯ e exp G x ¯ c , x ¯ m , x ¯ e ρ 1 exp G x ¯ c , x ¯ m , 0 ρ 1 = η e δ e ζ x ¯ c x ¯ e exp (G( bar(x)_(c), bar(x)_(m), bar(x)_(e)))^(rho-1)-exp (G( bar(x)_(c), bar(x)_(m),0))^(rho-1)=(eta_(e)-delta_(e)zeta( bar(x)_(c))) bar(x)_(e)\exp \left(G\left(\bar{x}_{c}, \bar{x}_{m}, \bar{x}_{e}\right)\right)^{\rho-1}-\exp \left(G\left(\bar{x}_{c}, \bar{x}_{m}, 0\right)\right)^{\rho-1}=\left(\eta_{e}-\delta_{e} \zeta\left(\bar{x}_{c}\right)\right) \bar{x}_{e}exp(G(x¯c,x¯m,x¯e))ρ1exp(G(x¯c,x¯m,0))ρ1=(ηeδeζ(x¯c))x¯e
which combined with equation (85) implies equation (36). We can also express Θ e Θ e Theta_(e)\Theta_{e}Θe as

where we used
s ¯ e = η e x ¯ e exp ( g ¯ Z ) ρ 1 s ¯ e = η e x ¯ e exp g ¯ Z ρ 1 bar(s)_(e)=(eta_(e) bar(x)_(e))/(exp ( bar(g)_(Z))^(rho-1))\bar{s}_{e}=\frac{\eta_{e} \bar{x}_{e}}{\exp \left(\bar{g}_{Z}\right)^{\rho-1}}s¯e=ηex¯eexp(g¯Z)ρ1
and
avsize c a s ¯ i z e e = ζ ( x ¯ c ) η e  avsize  c a s ¯ i z e e = ζ x ¯ c η e (" avsize "_(c))/(a_( bar(s)ize_(e)))=(zeta( bar(x)_(c)))/(eta_(e))\frac{\text { avsize }_{c}}{a_{\bar{s} i z e_{e}}}=\frac{\zeta\left(\bar{x}_{c}\right)}{\eta_{e}} avsize cas¯izee=ζ(x¯c)ηe
where a v s ¯ s i z e c a v s ¯ s i z e ¯ c av bar(s) bar(size)_(c)a v \bar{s} \overline{s i z e}_{c}avs¯sizec denotes the average size of continuing products and avsize denotes the average size of new products acquired by entering firms. Therefore, given a target for the term in square brackets in equation (36), and given data on a v s ¯ i z e c a v s ¯ i z e e a v s ¯ i z e c a v s ¯ i z e e (av( bar(s))ize_(c))/(av( bar(s))ize_(e))\frac{a v \bar{s} i z e_{c}}{a v \bar{s} i z e_{e}}avs¯izecavs¯izee and s ¯ e s ¯ e bar(s)_(e)\bar{s}_{e}s¯e, we can back out the implied level of δ e δ e delta_(e)\delta_{e}δe.
By equations (66) and (64),
(87) Θ c = [ 1 τ ¯ c 1 τ ¯ e ] [ 1 1 δ e a v s i z z c a v s ¯ i z e e ] Θ e = [ 1 τ ¯ c ρ 1 ] [ 1 + R ¯ exp ( g ¯ Y ) ] i ¯ r v ¯ (87) Θ c = 1 τ ¯ c 1 τ ¯ e 1 1 δ e a v s i z z c ¯ a v s ¯ i z e e Θ e = 1 τ ¯ c ρ 1 1 + R ¯ exp g ¯ Y i ¯ r v ¯ {:(87)Theta_(c)=[(1- bar(tau)_(c))/(1- bar(tau)_(e))][(1)/(1-delta_(e)(av bar(sizz_(c)))/(a_(v)( bar(s))ize_(e)))]Theta_(e)=[(1- bar(tau)_(c))/(rho-1)][(1+( bar(R)))/(exp( bar(g)_(Y)))]( bar(i)_(r))/( bar(v)^(')):}\begin{equation*} \Theta_{c}=\left[\frac{1-\bar{\tau}_{c}}{1-\bar{\tau}_{e}}\right]\left[\frac{1}{1-\delta_{e} \frac{a v \overline{s i z z_{c}}}{a_{v} \bar{s} i z e_{e}}}\right] \Theta_{e}=\left[\frac{1-\bar{\tau}_{c}}{\rho-1}\right]\left[\frac{1+\bar{R}}{\exp \left(\bar{g}_{Y}\right)}\right] \frac{\bar{i}_{r}}{\bar{v}^{\prime}} \tag{87} \end{equation*}(87)Θc=[1τ¯c1τ¯e][11δeavsizzcavs¯izee]Θe=[1τ¯cρ1][1+R¯exp(g¯Y)]i¯rv¯
where the second equality uses equations (39) and (86). Therefore, the value of Θ c Θ c Theta_(c)\Theta_{c}Θc is
independent of the extent of business stealing δ e , δ m δ e , δ m delta_(e),delta_(m)\delta_{e}, \delta_{m}δe,δm (or, similarly, is independent of the contribution of entrants to productivity growth).
Finally, by equations (67), (64), and (65),
(88) Θ m = [ 1 τ ¯ m 1 τ ¯ e ] [ 1 δ m a v s ¯ i z e c a v s ¯ i z e m 1 δ e a v s ¯ i z e m a v s ¯ i z e e ] Θ e = [ 1 τ ¯ m 1 τ ¯ c ] [ 1 δ m a v s ¯ i z e c avsize m ] Θ c (88) Θ m = 1 τ ¯ m 1 τ ¯ e 1 δ m a v s ¯ i z e c a v s ¯ i z e m 1 δ e a v s ¯ i z e m a v s ¯ i z e e Θ e = 1 τ ¯ m 1 τ ¯ c 1 δ m a v s ¯ i z e c ¯ avsize m Θ c {:(88)Theta_(m)=[(1- bar(tau)_(m))/(1- bar(tau)_(e))][(1-delta_(m)(av( bar(s))ize_(c))/(av( bar(s))ize_(m)))/(1-delta_(e)(av( bar(s))ize_(m))/(av( bar(s))ize_(e)))]Theta_(e)=[(1- bar(tau)_(m))/(1- bar(tau)_(c))][1-delta_(m)(av( bar(s)) bar(ize_(c)))/((avsize_(m)))]Theta_(c):}\begin{equation*} \Theta_{m}=\left[\frac{1-\bar{\tau}_{m}}{1-\bar{\tau}_{e}}\right]\left[\frac{1-\delta_{m} \frac{a v \bar{s} i z e_{c}}{a v \bar{s} i z e_{m}}}{1-\delta_{e} \frac{a v \bar{s} i z e_{m}}{a v \bar{s} i z e_{e}}}\right] \Theta_{e}=\left[\frac{1-\bar{\tau}_{m}}{1-\bar{\tau}_{c}}\right]\left[1-\delta_{m} \frac{a v \bar{s} \overline{i z e_{c}}}{\operatorname{avsize_{m}}}\right] \Theta_{c} \tag{88} \end{equation*}(88)Θm=[1τ¯m1τ¯e][1δmavs¯izecavs¯izem1δeavs¯izemavs¯izee]Θe=[1τ¯m1τ¯c][1δmavs¯izecavsizem]Θc
where the second equation uses equation (87).

C. 2 Data and calibration

We set the time period in the model to one year and calibrate the BGP of the model using average data for the period 1990-2014 (whenever possible).
We first describe how we assign values to the parameters determining the elasticities required to evaluate the aggregate implications of a change in innovation policies. We then describe how we parameterize the model to evaluate its full nonlinear dynamics.
List of model parameters to calculate analytic elasticities The parameters and BGP statistics that we need to assign to calculate our elasticities analytically are: the degree of intertemporal knowledge spillovers ϕ ϕ phi\phiϕ, the impact elasticity Θ Θ Theta\ThetaΘ, the share of physical capital in costs α α alpha\alphaα, the share of production labor in firms' output (inclusive of production subsidies) ( 1 α ) μ ( 1 α ) μ ((1-alpha))/(mu)\frac{(1-\alpha)}{\mu}(1α)μ, the share of innovative investment in firms' output (inclusive of production subsidies) i ¯ r i ¯ r bar(i)_(r)\bar{i}_{r}i¯r, the discount factor (which equals the ratio of the output growth rate to the interest rate, β ~ = exp ( g ¯ γ ) 1 + R ¯ ) β ~ = exp ( g ¯ γ ) 1 + R ¯ {:( tilde(beta))=(exp(( bar(g))gamma))/(1+( bar(R))))\left.\tilde{\beta}=\frac{\exp (\bar{g} \gamma)}{1+\bar{R}}\right)β~=exp(g¯γ)1+R¯), the parameter governing the intertemporal elasticity of substitution γ γ gamma\gammaγ, consumption-to-output ratio C ¯ Y ¯ C ¯ Y ¯ (( bar(C)))/(( bar(Y)))\frac{\bar{C}}{\bar{Y}}C¯Y¯, and the depreciation rate of physical capital d k d k d_(k)d_{k}dk (which is required to calculate the dynamics of the rental rate R k R k R_(k)R_{k}Rk ). We set Θ Θ Theta\ThetaΘ equal to its upper bound, Θ e Θ e Theta_(e)\Theta_{e}Θe. To measure Θ e Θ e Theta_(e)\Theta_{e}Θe, we use expression (36), which requires assigning values to the elasticity of substitution ρ ρ rho\rhoρ, the baseline productivity growth rate g ¯ Z g ¯ Z bar(g)_(Z)\bar{g}_{Z}g¯Z (which, given α α alpha\alphaα and g ¯ Y g ¯ Y bar(g)_(Y)\bar{g}_{Y}g¯Y, requires a value for the growth rate of the labor force g ¯ L g ¯ L bar(g)_(L)\bar{g}_{L}g¯L ), the counterfactual growth rate when investment by entrants is set to zero G ( x ¯ c , x ¯ m , 0 ) G x ¯ c , x ¯ m , 0 G( bar(x)_(c), bar(x)_(m),0)G\left(\bar{x}_{c}, \bar{x}_{m}, 0\right)G(x¯c,x¯m,0), and the ratio of innovative investments by entrants to total investments x ¯ e Y ¯ r x ¯ e Y ¯ r ( bar(x)_(e))/( bar(Y)_(r))\frac{\bar{x}_{e}}{\bar{Y}_{r}}x¯eY¯r. To measure x ¯ e Y ¯ r x ¯ e Y ¯ r ( bar(x)_(e))/( bar(Y)_(r))\frac{\bar{x}_{e}}{\bar{Y}_{r}}x¯eY¯r we use equation 39 (together with 40), which require measures of dividends relative to output d ¯ d ¯ bar(d)\bar{d}d¯, employment shares of entering firms s ¯ e s ¯ e bar(s)_(e)\bar{s}_{e}s¯e, and innovation subsidy rates for investment by entrants τ e τ e tau_(e)\tau_{e}τe. We also require initial BGP values for the innovation subsidies for incumbent firms, τ c τ c tau_(c)\tau_{c}τc and τ m τ m tau_(m)\tau_{m}τm, and the production subsidy (which we set to undo the distortions on capital accumulation arising from markups and the corporate profits tax). In measuring the rental rate of capital and firm dividends, we introduce a corporate profits tax, which we denote
by τ corp τ corp  tau_("corp ")\tau_{\text {corp }}τcorp . This tax is applied to the variable profits of intermediate good firms and the return to physical capital. In what follows, we describe how we assign values to these parameters and targets, which are reported in Table 2.
As described in the text, we consider two values of ϕ : ϕ = 1.6 ϕ : ϕ = 1.6 phi:phi=-1.6\phi: \phi=-1.6ϕ:ϕ=1.6 and ϕ = 0.96 ϕ = 0.96 phi=0.96\phi=0.96ϕ=0.96. We set the elasticity of substitution ρ ρ rho\rhoρ to 4 in the CES aggregator in equation (4). Note that this assumption fixes the term 1 / ( ρ 1 ) 1 / ( ρ 1 ) 1//(rho-1)1 /(\rho-1)1/(ρ1) that enters into our impact elasticity formulas. Holding the equilibrium markup μ μ mu\muμ fixed, changes in this parameter ρ ρ rho\rhoρ have no impact on any other implication of our model for the elasticities of aggregate productivity and welfare with respect to changes in aggregate innovative investment other than through this term. Hence, it is clear that our measured elasticities will be larger if we choose a value of 1 < ρ < 4 1 < ρ < 4 1 < rho < 41<\rho<41<ρ<4 and smaller if we choose a value of ρ > 4 ρ > 4 rho > 4\rho>4ρ>4. Note that our upper bound for the impact elasticity in equation (37) is independent of ρ ρ rho\rhoρ.
We fix a BGP consumption interest rate of R ¯ = 4 % R ¯ = 4 % bar(R)=4%\bar{R}=4 \%R¯=4%. This rate of return is close to that estimated by Poterba (1998) (5.1%), Hall (2003) (4.46%), and McGrattan and Prescott (2005) ( 4.1 % ( 4.1 % (4.1%(4.1 \%(4.1% for 1990-2001). We have set it at the low end of these estimates to reflect the decline in real interest rates that has occurred over the past several decades. We set the growth rate of the labor force g ¯ L = 0.007 g ¯ L = 0.007 bar(g)_(L)=0.007\bar{g}_{L}=0.007g¯L=0.007 corresponding to the annual growth rate of employment in the nonfinancial corporate sector over the period 1990-2014. We set the intertemporal elasticity parameter γ 1 γ 1 gamma rarr1\gamma \rightarrow 1γ1 (i.e., log log log\loglog utility). 27 27 ^(27){ }^{27}27
Output and innovative investment by incumbent firms To measure innovative investment and factor payments, we use data from the integrated macroeconomic accounts and fixed assets tables for the US nonfinancial corporate sector derived from the US national income and product accounts (NIPA).
Since 2013, NIPA includes measures of investments and stocks in intellectual property products going back to 1929. These intellectual property products include research and development, software, and artistic and literary originals. Research and development by the nonfinancial corporate sector accounts for roughly half of these investments.
In our model, total expenditures on innovative investments correspond to output of the research good P r t Y r t P r t Y r t P_(rt)Y_(rt)P_{r t} Y_{r t}PrtYrt, which equals the compensation of research labor W t L r t W t L r t W_(t)L_(rt)W_{t} L_{r t}WtLrt. We presume that what is measured in the data is the innovative investment expenditures of incumbent firms P r t ( x c t + x m t ) P r t x c t + x m t P_(rt)(x_(ct)+x_(mt))P_{r t}\left(x_{c t}+x_{m t}\right)Prt(xct+xmt), and infer innovative investment expenditures of entering
firms P r t x e t P r t x e t P_(rt)x_(et)P_{r t} x_{e t}Prtxet (since we assume that entrants are not in the nonfinancial corporate sector when they make these expenditures).
We measure aggregate firms' output inclusive of the production subsidy in our model, ( 1 + τ y ) Y t 1 + τ y Y t (1+tau_(y))Y_(t)\left(1+\tau_{y}\right) Y_{t}(1+τy)Yt, as gross value added of the nonfinancial corporate sector less indirect business taxes less measured innovative investment expenditures by incumbent firms. With these adjustments, we obtain P ¯ r t ( x ¯ c + x ¯ m ) ( 1 + τ y ) Y ¯ t = 0.061 P ¯ r t x ¯ c + x ¯ m 1 + τ y Y ¯ t = 0.061 ( bar(P)_(rt)( bar(x)_(c)+ bar(x)_(m)))/((1+tau_(y)) bar(Y)_(t))=0.061\frac{\bar{P}_{r t}\left(\bar{x}_{c}+\bar{x}_{m}\right)}{\left(1+\tau_{y}\right) \bar{Y}_{t}}=0.061P¯rt(x¯c+x¯m)(1+τy)Y¯t=0.061 (the average of this measure for the period 1990-2014). The corresponding BGP growth rate of output of the final consumption good is g ¯ Y = 0.025 g ¯ Y = 0.025 bar(g)_(Y)=0.025\bar{g}_{Y}=0.025g¯Y=0.025 and the discount factor β ~ = exp ( g ¯ Y ) / ( 1 + R ¯ ) = 0.986 β ~ = exp g ¯ Y / ( 1 + R ¯ ) = 0.986 tilde(beta)=exp( bar(g)_(Y))//(1+ bar(R))=0.986\tilde{\beta}=\exp \left(\bar{g}_{Y}\right) /(1+\bar{R})=0.986β~=exp(g¯Y)/(1+R¯)=0.986.
Payments to physical capital and profits In the data, gross value added of the nonfinancial corporate sector is decomposed into indirect business taxes, compensation of employees, and gross operating surplus. We measure compensation of production labor W t L p t W t L p t W_(t)L_(pt)W_{t} L_{p t}WtLpt in our model as compensation of employees in the nonfinancial corporate sector less measured innovative investment expenditures by incumbent firms.
In our model, a portion of gross operating surplus is paid as rental payments to physical capital, a portion is paid as corporate profits taxes (taxes on income and wealth in the NIPA), and a portion is paid to owners of firms as after-tax variable profits less expenditures on innovative investment. To measure rental payments to physical capital, R k K t R k K t R_(k)K_(t)R_{k} K_{t}RkKt, we impute the rental rate to physical capital using the standard user cost formula implied by our model, R ¯ k = R ¯ 1 τ c o r p + d k R ¯ k = R ¯ 1 τ c o r p + d k bar(R)_(k)=(( bar(R)))/(1-tau_(corp))+d_(k)\bar{R}_{k}=\frac{\bar{R}}{1-\tau_{c o r p}}+d_{k}R¯k=R¯1τcorp+dk, with a depreciation rate of physical capital d k = 0.055 d k = 0.055 d_(k)=0.055d_{k}=0.055dk=0.055 and a corporate profits tax τ c o r p = 0.161 τ c o r p = 0.161 tau_(corp)=0.161\tau_{c o r p}=0.161τcorp=0.161 calculated as described below, which imply R ¯ k = 0.1027 R ¯ k = 0.1027 bar(R)_(k)=0.1027\bar{R}_{k}=0.1027R¯k=0.1027. We measure K t K t K_(t)K_{t}Kt by the replacement value of physical capital ("Nonfinancial assets, nonfinancial corporate business") minus the replacement value of intangible capital ("Nonresidential intellectual property products, nonfinancial corporate business, current cost basis"), which results in K ¯ t = 2.125 × ( 1 + τ y ) Y ¯ t K ¯ t = 2.125 × 1 + τ y Y ¯ t bar(K)_(t)=2.125 xx(1+tau_(y)) bar(Y)_(t)\bar{K}_{t}=2.125 \times\left(1+\tau_{y}\right) \bar{Y}_{t}K¯t=2.125×(1+τy)Y¯t. We measure the depreciation rate d k d k d_(k)d_{k}dk as the ratio of "Consumption of fixed capital, structures, equipment, and intellectual property products, nonfinancial corporate business" minus consumption of intangible capital, to the replacement value of physical capital. In order to measure consumption of intangible capital, we multiply its replacement value by its depreciation rate. We measure the corporate profits tax τ c o r p τ c o r p tau_(corp)\tau_{c o r p}τcorp as the ratio of payments of taxes on income and wealth in the data, 0.037 × ( 1 + τ y ) Y ¯ t 0.037 × 1 + τ y Y ¯ t 0.037 xx(1+tau_(y)) bar(Y)_(t)0.037 \times\left(1+\tau_{y}\right) \bar{Y}_{t}0.037×(1+τy)Y¯t, to the tax base of this tax in our model (i.e., variable profits plus physical capital income net of depreciation, ( 1 + τ y ) Y ¯ t W ¯ t L ¯ p t d k K ¯ t = 0.228 × ( 1 + τ y ) Y ¯ t ) 1 + τ y Y ¯ t W ¯ t L ¯ p t d k K ¯ t = 0.228 × 1 + τ y Y ¯ t {:(1+tau_(y)) bar(Y)_(t)- bar(W)_(t) bar(L)_(pt)-d_(k) bar(K)_(t)=0.228 xx(1+tau_(y)) bar(Y)_(t))\left.\left(1+\tau_{y}\right) \bar{Y}_{t}-\bar{W}_{t} \bar{L}_{p t}-d_{k} \bar{K}_{t}=0.228 \times\left(1+\tau_{y}\right) \bar{Y}_{t}\right)(1+τy)Y¯tW¯tL¯ptdkK¯t=0.228×(1+τy)Y¯t). In choosing this tax base, we have assumed that firms cannot expense their investments in innovation from profits for tax purposes. We make this assumption so that it is possible for the equilibrium to be conditionally efficient in the presence of uniform innovation subsidies and a positive corporate
profits tax.
Following this procedure, we obtain an average share (over the period 1990-2014) of production labor in firms' output (inclusive of production subsidies) of W ¯ t L ¯ p t ( 1 + τ ¯ y ) Y ¯ t = W ¯ t L ¯ p t 1 + τ ¯ y Y ¯ t = ( bar(W)_(t) bar(L)_(pt))/((1+ bar(tau)_(y)) bar(Y)_(t))=\frac{\bar{W}_{t} \bar{L}_{p t}}{\left(1+\bar{\tau}_{y}\right) \bar{Y}_{t}}=W¯tL¯pt(1+τ¯y)Y¯t= ( 1 α ) μ = 0.654 ( 1 α ) μ = 0.654 ((1-alpha))/(mu)=0.654\frac{(1-\alpha)}{\mu}=0.654(1α)μ=0.654, a share of capital R ¯ t K ¯ t ( 1 + τ ¯ y ) Y ¯ t = α μ = 0.218 R ¯ t K ¯ t 1 + τ ¯ y Y ¯ t = α μ = 0.218 ( bar(R)_(t) bar(K)_(t))/((1+ bar(tau)_(y)) bar(Y)_(t))=(alpha )/(mu)=0.218\frac{\bar{R}_{t} \bar{K}_{t}}{\left(1+\bar{\tau}_{y}\right) \bar{Y}_{t}}=\frac{\alpha}{\mu}=0.218R¯tK¯t(1+τ¯y)Y¯t=αμ=0.218, and a share of variable profits in output ( 1 R ¯ k t K ¯ t + W ¯ t L ¯ p t ( 1 + τ ¯ y ) Y ¯ t ) = ( μ 1 μ ) = 0.127 1 R ¯ k t K ¯ t + W ¯ t L ¯ p t 1 + τ ¯ y Y ¯ t = μ 1 μ = 0.127 (1-( bar(R)_(kt) bar(K)_(t)+ bar(W)_(t) bar(L)_(pt))/((1+ bar(tau)_(y)) bar(Y)_(t)))=((mu-1)/(mu))=0.127\left(1-\frac{\bar{R}_{k t} \bar{K}_{t}+\bar{W}_{t} \bar{L}_{p t}}{\left(1+\bar{\tau}_{y}\right) \bar{Y}_{t}}\right)=\left(\frac{\mu-1}{\mu}\right)=0.127(1R¯ktK¯t+W¯tL¯pt(1+τ¯y)Y¯t)=(μ1μ)=0.127, so α = 0.25 α = 0.25 alpha=0.25\alpha=0.25α=0.25 and μ = 1.146 μ = 1.146 mu=1.146\mu=1.146μ=1.146. We assume that in the initial BGP innovative subsidies are uniform, with τ ¯ e = τ ¯ c = τ ¯ m = 0.03 τ ¯ e = τ ¯ c = τ ¯ m = 0.03 bar(tau)_(e)= bar(tau)_(c)= bar(tau)_(m)=0.03\bar{\tau}_{e}=\bar{\tau}_{c}=\bar{\tau}_{m}=0.03τ¯e=τ¯c=τ¯m=0.03, obtained from Tyson and Linden (2012) (Table 7 in that paper reports estimates of corporate R&D tax credit claims relative to R&D business spending in the United States). This gives us
d ¯ = D ¯ t ( 1 + τ y ) Y ¯ t = ( μ 1 μ ) ( 1 τ ¯ corp ) ( 1 τ ¯ e ) P ¯ r t ( x ¯ c + x ¯ m ) ( 1 + τ ¯ y ) Y ¯ t = 0.047 d ¯ = D ¯ t 1 + τ y Y ¯ t = μ 1 μ 1 τ ¯ corp  1 τ ¯ e P ¯ r t x ¯ c + x ¯ m 1 + τ ¯ y Y ¯ t = 0.047 bar(d)=( bar(D)_(t))/((1+tau_(y)) bar(Y)_(t))=((mu-1)/(mu))(1- bar(tau)_("corp "))-(1- bar(tau)_(e))( bar(P)_(rt)( bar(x)_(c)+ bar(x)_(m)))/((1+ bar(tau)_(y)) bar(Y)_(t))=0.047\bar{d}=\frac{\bar{D}_{t}}{\left(1+\tau_{y}\right) \bar{Y}_{t}}=\left(\frac{\mu-1}{\mu}\right)\left(1-\bar{\tau}_{\text {corp }}\right)-\left(1-\bar{\tau}_{e}\right) \frac{\bar{P}_{r t}\left(\bar{x}_{c}+\bar{x}_{m}\right)}{\left(1+\bar{\tau}_{y}\right) \bar{Y}_{t}}=0.047d¯=D¯t(1+τy)Y¯t=(μ1μ)(1τ¯corp )(1τ¯e)P¯rt(x¯c+x¯m)(1+τ¯y)Y¯t=0.047
In order to remove the distortions induced by the markup and the corporate profits tax in the allocation of physical capital when calculating welfare, we set the production subsidy τ ¯ y = μ R ¯ + d k ( R ¯ ( 1 τ corp ) + d k ) 1 = 0.24 . 28 τ ¯ y = μ R ¯ + d k R ¯ 1 τ corp  + d k 1 = 0.24 . 28 bar(tau)_(y)=(mu)/(( bar(R))+d_(k))((( bar(R)))/((1-tau_("corp ")))+d_(k))-1=0.24.^(28)\bar{\tau}_{y}=\frac{\mu}{\bar{R}+d_{k}}\left(\frac{\bar{R}}{\left(1-\tau_{\text {corp }}\right)}+d_{k}\right)-1=0.24 .{ }^{28}τ¯y=μR¯+dk(R¯(1τcorp )+dk)1=0.24.28 Our choice of τ ¯ y τ ¯ y bar(tau)_(y)\bar{\tau}_{y}τ¯y only affects our welfare calculations. Given measures of α , g ¯ L α , g ¯ L alpha, bar(g)_(L)\alpha, \bar{g}_{L}α,g¯L, and g ¯ Y g ¯ Y bar(g)_(Y)\bar{g}_{Y}g¯Y, we construct estimates of the BGP growth rate of total factor productivity g ¯ Z = 0.0136 g ¯ Z = 0.0136 bar(g)_(Z)=0.0136\bar{g}_{Z}=0.0136g¯Z=0.0136. We calculate the ratio of consumption to output as
C ¯ Y ¯ = 1 I ¯ K ¯ K ¯ Y ¯ = 1 ( exp ( g ¯ Y ) ( 1 d k ) ) α ( 1 + τ y ) μ R ¯ k = 0.789 C ¯ Y ¯ = 1 I ¯ K ¯ K ¯ Y ¯ ¯ = 1 exp g ¯ Y 1 d k α 1 + τ y μ R ¯ k = 0.789 (( bar(C)))/(( bar(Y)))=1-(( bar(I)))/(( bar(K))) bar(K) bar(bar(Y))=1-(exp( bar(g)_(Y))-(1-d_(k)))((alpha(1+tau_(y)))/(mu bar(R)_(k)))=0.789\frac{\bar{C}}{\bar{Y}}=1-\frac{\bar{I}}{\bar{K}} \bar{K} \overline{\bar{Y}}=1-\left(\exp \left(\bar{g}_{Y}\right)-\left(1-d_{k}\right)\right) \frac{\alpha\left(1+\tau_{y}\right)}{\mu \bar{R}_{k}}=0.789C¯Y¯=1I¯K¯K¯Y¯=1(exp(g¯Y)(1dk))α(1+τy)μR¯k=0.789
Product-level dynamics To measure the employment shares of entering firms s ¯ e s ¯ e bar(s)_(e)\bar{s}_{e}s¯e, we use annual data from the Longitudinal Business Database (LBD) in the United States on the dynamics of establishments and firms that own them. We make the identifying assumption that an establishment in the LBD data corresponds to an intermediate good in the model. Products in entering firms in the model correspond to establishments in the data that are new (they are active in year t t ttt but not in year t 1 t 1 t-1t-1t1 ) and are owned by new firms. The average value of s ¯ e s ¯ e bar(s)_(e)\bar{s}_{e}s¯e in the period 1990-2014 is 0.027 . If we assume time periods that are longer than one year, entrants represent a higher share in the total number of establishments and in employment. When we solve the model nonlinearly, we calibrate other parameter values using additional information on product-level dynamics.
By equation (40), with d ¯ = 0.047 d ¯ = 0.047 bar(d)=0.047\bar{d}=0.047d¯=0.047, exp ( g ¯ Y ) / ( 1 + R ¯ ) = 0.986 exp g ¯ Y / ( 1 + R ¯ ) = 0.986 exp( bar(g)_(Y))//(1+ bar(R))=0.986\exp \left(\bar{g}_{Y}\right) /(1+\bar{R})=0.986exp(g¯Y)/(1+R¯)=0.986, and s ¯ e = 0.027 s ¯ e = 0.027 bar(s)_(e)=0.027\bar{s}_{e}=0.027s¯e=0.027, we obtain v ¯ = 1.15 v ¯ = 1.15 bar(v)=1.15\bar{v}=1.15v¯=1.15. By equation (39), we obtain P ¯ r t x ¯ e ( 1 + τ y ) Y ¯ t = 0.031 P ¯ r t x ¯ e 1 + τ y Y ¯ t = 0.031 ( bar(P)_(rt) bar(x)_(e))/((1+tau_(y)) bar(Y)_(t))=0.031\frac{\bar{P}_{r t} \bar{x}_{e}}{\left(1+\tau_{y}\right) \bar{Y}_{t}}=0.031P¯rtx¯e(1+τy)Y¯t=0.031, so i ¯ r = P ¯ r t ( x ¯ e + x ¯ c + x ¯ m ) ( 1 + τ y ) Y ¯ t = i ¯ r = P ¯ r t x ¯ e + x ¯ c + x ¯ m 1 + τ y Y ¯ t = bar(i)_(r)=( bar(P)_(rt)( bar(x)_(e)+ bar(x)_(c)+ bar(x)_(m)))/((1+tau_(y)) bar(Y)_(t))=\bar{i}_{r}=\frac{\bar{P}_{r t}\left(\bar{x}_{e}+\bar{x}_{c}+\bar{x}_{m}\right)}{\left(1+\tau_{y}\right) \bar{Y}_{t}}=i¯r=P¯rt(x¯e+x¯c+x¯m)(1+τy)Y¯t=
0.061 + 0.031 = 0.093 0.061 + 0.031 = 0.093 0.061+0.031=0.0930.061+0.031=0.0930.061+0.031=0.093. The fiscal cost on innovation spending, E / Y E / Y E//YE / YE/Y, in the initial BGP is given by 0.03 × i ¯ r = 0.0028 0.03 × i ¯ r = 0.0028 0.03 xx bar(i)_(r)=0.00280.03 \times \bar{i}_{r}=0.00280.03×i¯r=0.0028. The ratio x ¯ e / Y ¯ r x ¯ e / Y ¯ r bar(x)_(e)// bar(Y)_(r)\bar{x}_{e} / \bar{Y}_{r}x¯e/Y¯r, which is required to calculate Θ e Θ e Theta_(e)\Theta_{e}Θe in equation (36), is equal to 0.032 / 0.093 = 0.34 0.032 / 0.093 = 0.34 0.032//0.093=0.340.032 / 0.093=0.340.032/0.093=0.34.
Business stealing Based on the derivation of equation (86) in Appendix B, the expression in square brackets in equation (36) can be written as
exp ( g ¯ Z ) ρ 1 exp ( G ( x ¯ c , x ¯ m , 0 ) ) ρ 1 exp ( g ¯ Z ) ρ 1 = ( 1 δ e a v . s ¯ i z e c a v . s i z e e ) s ¯ e exp g ¯ Z ρ 1 exp G x ¯ c , x ¯ m , 0 ρ 1 exp g ¯ Z ρ 1 = 1 δ e a v . s ¯ i z e c a v . s i z e e ¯ s ¯ e (exp ( bar(g)_(Z))^(rho-1)-exp (G( bar(x)_(c), bar(x)_(m),0))^(rho-1))/(exp ( bar(g)_(Z))^(rho-1))=(1-delta_(e)(av.( bar(s))ize_(c))/(av. bar(size_(e)))) bar(s)_(e)\frac{\exp \left(\bar{g}_{Z}\right)^{\rho-1}-\exp \left(G\left(\bar{x}_{c}, \bar{x}_{m}, 0\right)\right)^{\rho-1}}{\exp \left(\bar{g}_{Z}\right)^{\rho-1}}=\left(1-\delta_{e} \frac{a v . \bar{s} i z e_{c}}{a v . \overline{s i z e_{e}}}\right) \bar{s}_{e}exp(g¯Z)ρ1exp(G(x¯c,x¯m,0))ρ1exp(g¯Z)ρ1=(1δeav.s¯izecav.sizee)s¯e
We consider two parameterizations of the extent of business stealing for entering firms. First, we set δ e = 0 δ e = 0 delta_(e)=0\delta_{e}=0δe=0, so that the expression in square brackets in equation (36) is equal to s ¯ e = 0.027 s ¯ e = 0.027 bar(s)_(e)=0.027\bar{s}_{e}=0.027s¯e=0.027. Second, we let δ e > 0 δ e > 0 delta_(e) > 0\delta_{e}>0δe>0, and choose G ( x ¯ c , x ¯ m , 0 ) G x ¯ c , x ¯ m , 0 G( bar(x)_(c), bar(x)_(m),0)G\left(\bar{x}_{c}, \bar{x}_{m}, 0\right)G(x¯c,x¯m,0) so that ( g ¯ Z G ( x ¯ c , x ¯ m , 0 ) ) / g ¯ Z ) = g ¯ Z G x ¯ c , x ¯ m , 0 / g ¯ Z = {:( bar(g)_(Z)-G( bar(x)_(c), bar(x)_(m),0))// bar(g)_(Z))=\left.\left(\bar{g}_{Z}-G\left(\bar{x}_{c}, \bar{x}_{m}, 0\right)\right) / \bar{g}_{Z}\right)=(g¯ZG(x¯c,x¯m,0))/g¯Z)= 0.257 , which corresponds to estimates of the portion of annual trend productivity growth due to entry in Akcigit and Kerr (2018). Given our estimate of g ¯ Z = 0.014 g ¯ Z = 0.014 bar(g)_(Z)=0.014\bar{g}_{Z}=0.014g¯Z=0.014 and ρ = 4 ρ = 4 rho=4\rho=4ρ=4, we obtain G ( x ¯ c , x ¯ m , 0 ) = 0.01 G x ¯ c , x ¯ m , 0 = 0.01 G( bar(x)_(c), bar(x)_(m),0)=0.01G\left(\bar{x}_{c}, \bar{x}_{m}, 0\right)=0.01G(x¯c,x¯m,0)=0.01 and the expression in square brackets in equation (36) is equal to 0.011. Given our measure a v s z i z e c a v s i z e e = 3 a v s z i z ¯ e c a v s i z ¯ e e = 3 (avs bar(ziz)e_(c))/(av bar(siz)e_(e))=3\frac{a v s \overline{z i z} e_{c}}{a v \overline{s i z} e_{e}}=3avszizecavsizee=3 discussed below, our parameterization implies δ e = 0.20 δ e = 0.20 delta_(e)=0.20\delta_{e}=0.20δe=0.20

C. 3 Calibration of additional parameters to solve nonlinear transition dynamics

In the previous section, we described how we assign values to the parameters determining the elasticities required to evaluate the aggregate implications of a change in innovation policies. We now discuss how we assign remaining parameter values to solve for all BGP variables and to solve the transition dynamics of the model nonlinearly and globally (in which we do not impose that Θ Θ Theta\ThetaΘ is at its upper bound of Θ e Θ e Theta_(e)\Theta_{e}Θe in response to proportional policy changes). These parameter values and targets are reported in Table 3.
In Section C. 2 we described how we choose the value of δ e δ e delta_(e)\delta_{e}δe. Here we impose δ m = δ e δ m = δ e delta_(m)=delta_(e)\delta_{m}=\delta_{e}δm=δe. Moreover, we specify
h ( x m ) = h 1 x m h 2 ζ ( x c ) = ζ 0 + ζ 1 x c ζ 2 h x m = h 1 x m h 2 ζ x c = ζ 0 + ζ 1 x c ζ 2 {:[h(x_(m))=h_(1)x_(m)^(h_(2))],[zeta(x_(c))=zeta_(0)+zeta_(1)x_(c)^(zeta_(2))]:}\begin{gathered} h\left(x_{m}\right)=h_{1} x_{m}^{h_{2}} \\ \zeta\left(x_{c}\right)=\zeta_{0}+\zeta_{1} x_{c}^{\zeta_{2}} \end{gathered}h(xm)=h1xmh2ζ(xc)=ζ0+ζ1xcζ2
where h 1 > 0 , ζ 1 > 0 , 0 < h 2 < 1 h 1 > 0 , ζ 1 > 0 , 0 < h 2 < 1 h_(1) > 0,zeta_(1) > 0,0 < h_(2) < 1h_{1}>0, \zeta_{1}>0,0<h_{2}<1h1>0,ζ1>0,0<h2<1, and 0 < ζ 2 < 1 0 < ζ 2 < 1 0 < zeta_(2) < 10<\zeta_{2}<10<ζ2<1. The parameters that need to be assigned are { δ 0 , η e , η m , h 1 , h 2 , ζ 0 , ζ 1 , ζ 2 } δ 0 , η e , η m , h 1 , h 2 , ζ 0 , ζ 1 , ζ 2 {delta_(0),eta_(e),eta_(m),h_(1),h_(2),zeta_(0),zeta_(1),zeta_(2)}\left\{\delta_{0}, \eta_{e}, \eta_{m}, h_{1}, h_{2}, \zeta_{0}, \zeta_{1}, \zeta_{2}\right\}{δ0,ηe,ηm,h1,h2,ζ0,ζ1,ζ2}.
Suppose we have data on the growth rate of the measure of products g M t = log ( M t + 1 / M t ) g M t = log M t + 1 / M t g_(Mt)=log(M_(t+1)//M_(t))g_{M t}=\log \left(M_{t+1} / M_{t}\right)gMt=log(Mt+1/Mt), as well as data on the fraction of products that are continuing products in incumbent firms
Shares in firms' output, Y × ( 1 + τ y ) Y × 1 + τ y Y xx(1+tau_(y))Y \times\left(1+\tau_{y}\right)Y×(1+τy) Parameters
Production labor compensation, W L p W L p WL_(p)W L_{p}WLp 0.655 Elasticity of substitution, ρ ρ rho\rhoρ 4
Physical capital compensation, R k K R k K R_(k)KR_{k} KRkK 0.218 Markup, μ μ mu\muμ 1.146
Variable profits, Π = ( 1 + τ y ) Y W L p R k K Π = 1 + τ y Y W L p R k K Pi=(1+tau_(y))Y-WL_(p)-R_(k)K\Pi=\left(1+\tau_{y}\right) Y-W L_{p}-R_{k} KΠ=(1+τy)YWLpRkK 0.127 Physical capital share in costs, α α alpha\alphaα 0.25
Innovation investments by incumbents, P r ( x c + x m ) P r x c + x m P_(r)(x_(c)+x_(m))P_{r}\left(x_{c}+x_{m}\right)Pr(xc+xm) 0.061 Intertemporal spillovers, ϕ ϕ phi\phiϕ -1.6 or 0.99
Dividends, ( 1 τ corr ) Π P r ( ( 1 τ c ) x c + ( 1 τ m ) x m ) 1 τ corr  Π P r 1 τ c x c + 1 τ m x m (1-tau_("corr "))Pi-P_(r)((1-tau_(c))x_(c)+(1-tau_(m))x_(m))\left(1-\tau_{\text {corr }}\right) \Pi-P_{r}\left(\left(1-\tau_{c}\right) x_{c}+\left(1-\tau_{m}\right) x_{m}\right)(1τcorr )ΠPr((1τc)xc+(1τm)xm) 0.047 Capital depreciation rate, d k d k d_(k)d_{k}dk 0.055
Innovation investments by entrants, P r x e P r x e P_(r)x_(e)P_{r} x_{e}Prxe 0.031 Interest rate, R ¯ R ¯ bar(R)\bar{R}R¯ 0.04
Innovation intensity, P r ( x c + x m + x e ) = P r Y r P r x c + x m + x e = P r Y r P_(r)(x_(c)+x_(m)+x_(e))=P_(r)Y_(r)P_{r}\left(x_{c}+x_{m}+x_{e}\right)=P_{r} Y_{r}Pr(xc+xm+xe)=PrYr 0.093 Intertemporal substitution, γ γ gamma\gammaγ 1
Implied allocation of labor, L p / L L p / L L_(p)//LL_{p} / LLp/L 0.876 Labor growth rate, g ¯ L g ¯ L bar(g)_(L)\bar{g}_{L}g¯L 0.007
Scientific progress growth, δ ¯ A r δ ¯ A r bar(delta)_(A_(r))\bar{\delta}_{A_{r}}δ¯Ar 0.029 or -0.006
Business stealing by entrants, δ e δ e delta_(e)\delta_{e}δe 0 or 0.20
Policies in initial BGP Other targets
Innovation subsidies, τ c = τ m = τ e τ c = τ m = τ e tau_(c)=tau_(m)=tau_(e)\tau_{c}=\tau_{m}=\tau_{e}τc=τm=τe 0.03 Output growth rate, g ¯ Y g ¯ Y bar(g)_(Y)\bar{g}_{Y}g¯Y 0.025
Output subsidy, τ y τ y tau_(y)\tau_{y}τy 0.24 Employment share of entrants, s ¯ e s ¯ e bar(s)_(e)\bar{s}_{e}s¯e 0.027
Corporate profit tax rate, τ corp τ corp  tau_("corp ")\tau_{\text {corp }}τcorp  0.16
0.257
Other model outcomes
Aggregate productivity growth, g ¯ Z g ¯ Z bar(g)_(Z)\bar{g}_{Z}g¯Z 0.014 Value incumbents relative to Y ¯ ( 1 + τ y ) Y ¯ 1 + τ y bar(Y)(1+tau_(y))\bar{Y}\left(1+\tau_{y}\right)Y¯(1+τy) 1.16
Discount factor, β ~ β ~ tilde(beta)\tilde{\beta}β~ 0.986 Term exp ( z ¯ z ) ρ 1 exp ( G ( x ¯ c , x ¯ 2 , 0 ) ) ρ 1 exp ( z ¯ ) p 1 exp ( z ¯ z ¯ ) ρ 1 exp G x ¯ c , x ¯ 2 , 0 ρ 1 exp ( z ¯ ) p 1 (exp( bar(( bar(z))z))^(rho-1)-exp (G( bar(x)_(c), bar(x)^(2),0))^(rho-1))/(exp(( bar(z)))^(p-1))\frac{\exp (\overline{\bar{z} z})^{\rho-1}-\exp \left(G\left(\bar{x}_{c}, \bar{x}^{2}, 0\right)\right)^{\rho-1}}{\exp (\bar{z})^{p-1}}exp(z¯z)ρ1exp(G(x¯c,x¯2,0))ρ1exp(z¯)p1 0.027 or 0.01
Consumption/output, C ¯ / Y ¯ C ¯ / Y ¯ bar(C)// bar(Y)\bar{C} / \bar{Y}C¯/Y¯ 0.789 Impact elasticity entry, Θ e Θ e Theta_(e)\Theta_{e}Θe 0.026 or 0.010
Shares in firms' output, Y xx(1+tau_(y)) Parameters Production labor compensation, WL_(p) 0.655 Elasticity of substitution, rho 4 Physical capital compensation, R_(k)K 0.218 Markup, mu 1.146 Variable profits, Pi=(1+tau_(y))Y-WL_(p)-R_(k)K 0.127 Physical capital share in costs, alpha 0.25 Innovation investments by incumbents, P_(r)(x_(c)+x_(m)) 0.061 Intertemporal spillovers, phi -1.6 or 0.99 Dividends, (1-tau_("corr "))Pi-P_(r)((1-tau_(c))x_(c)+(1-tau_(m))x_(m)) 0.047 Capital depreciation rate, d_(k) 0.055 Innovation investments by entrants, P_(r)x_(e) 0.031 Interest rate, bar(R) 0.04 Innovation intensity, P_(r)(x_(c)+x_(m)+x_(e))=P_(r)Y_(r) 0.093 Intertemporal substitution, gamma 1 Implied allocation of labor, L_(p)//L 0.876 Labor growth rate, bar(g)_(L) 0.007 Scientific progress growth, bar(delta)_(A_(r)) 0.029 or -0.006 Business stealing by entrants, delta_(e) 0 or 0.20 Policies in initial BGP Other targets Innovation subsidies, tau_(c)=tau_(m)=tau_(e) 0.03 Output growth rate, bar(g)_(Y) 0.025 Output subsidy, tau_(y) 0.24 Employment share of entrants, bar(s)_(e) 0.027 Corporate profit tax rate, tau_("corp ") 0.16 https://cdn.mathpix.com/cropped/2024_08_28_d2b2a6121bfaf1583836g-08.jpg?height=65&width=577&top_left_y=1578&top_left_x=1218 0.257 Other model outcomes Aggregate productivity growth, bar(g)_(Z) 0.014 Value incumbents relative to bar(Y)(1+tau_(y)) 1.16 Discount factor, tilde(beta) 0.986 Term (exp( bar(( bar(z))z))^(rho-1)-exp (G( bar(x)_(c), bar(x)^(2),0))^(rho-1))/(exp(( bar(z)))^(p-1)) 0.027 or 0.01 Consumption/output, bar(C)// bar(Y) 0.789 Impact elasticity entry, Theta_(e) 0.026 or 0.010| Shares in firms' output, $Y \times\left(1+\tau_{y}\right)$ | | Parameters | | | :---: | :---: | :---: | :---: | | Production labor compensation, $W L_{p}$ | 0.655 | Elasticity of substitution, $\rho$ | 4 | | Physical capital compensation, $R_{k} K$ | 0.218 | Markup, $\mu$ | 1.146 | | Variable profits, $\Pi=\left(1+\tau_{y}\right) Y-W L_{p}-R_{k} K$ | 0.127 | Physical capital share in costs, $\alpha$ | 0.25 | | Innovation investments by incumbents, $P_{r}\left(x_{c}+x_{m}\right)$ | 0.061 | Intertemporal spillovers, $\phi$ | -1.6 or 0.99 | | Dividends, $\left(1-\tau_{\text {corr }}\right) \Pi-P_{r}\left(\left(1-\tau_{c}\right) x_{c}+\left(1-\tau_{m}\right) x_{m}\right)$ | 0.047 | Capital depreciation rate, $d_{k}$ | 0.055 | | Innovation investments by entrants, $P_{r} x_{e}$ | 0.031 | Interest rate, $\bar{R}$ | 0.04 | | Innovation intensity, $P_{r}\left(x_{c}+x_{m}+x_{e}\right)=P_{r} Y_{r}$ | 0.093 | Intertemporal substitution, $\gamma$ | 1 | | Implied allocation of labor, $L_{p} / L$ | 0.876 | Labor growth rate, $\bar{g}_{L}$ | 0.007 | | | | Scientific progress growth, $\bar{\delta}_{A_{r}}$ | 0.029 or -0.006 | | | | Business stealing by entrants, $\delta_{e}$ | 0 or 0.20 | | Policies in initial BGP | | Other targets | | | Innovation subsidies, $\tau_{c}=\tau_{m}=\tau_{e}$ | 0.03 | Output growth rate, $\bar{g}_{Y}$ | 0.025 | | Output subsidy, $\tau_{y}$ | 0.24 | Employment share of entrants, $\bar{s}_{e}$ | 0.027 | | Corporate profit tax rate, $\tau_{\text {corp }}$ | 0.16 | ![](https://cdn.mathpix.com/cropped/2024_08_28_d2b2a6121bfaf1583836g-08.jpg?height=65&width=577&top_left_y=1578&top_left_x=1218) | 0.257 | | Other model outcomes | | | | | Aggregate productivity growth, $\bar{g}_{Z}$ | 0.014 | Value incumbents relative to $\bar{Y}\left(1+\tau_{y}\right)$ | 1.16 | | Discount factor, $\tilde{\beta}$ | 0.986 | Term $\frac{\exp (\overline{\bar{z} z})^{\rho-1}-\exp \left(G\left(\bar{x}_{c}, \bar{x}^{2}, 0\right)\right)^{\rho-1}}{\exp (\bar{z})^{p-1}}$ | 0.027 or 0.01 | | Consumption/output, $\bar{C} / \bar{Y}$ | 0.789 | Impact elasticity entry, $\Theta_{e}$ | 0.026 or 0.010 |
Table 2: Baseline parameters, targets, and outcomes
f c t + 1 f c t + 1 f_(ct+1)f_{c t+1}fct+1, the fraction of products that are new to incumbent firms measured as the sum of those that are new to society and stolen f m t + 1 f m t + 1 f_(mt+1)f_{m t+1}fmt+1, and the fraction of products that are produced in entering firms measured as the sum of those that are new to society and stolen f e t + 1 = 1 f c t + 1 f m t + 1 f e t + 1 = 1 f c t + 1 f m t + 1 f_(et+1)=1-f_(ct+1)-f_(mt+1)f_{e t+1}=1-f_{c t+1}-f_{m t+1}fet+1=1fct+1fmt+1. Suppose we also have data on the aggregate size of continuing products in incumbent firms s c t + 1 s c t + 1 s_(ct+1)s_{c t+1}sct+1, the aggregate size of products that are new to incumbent firms measured as the sum of those that are new products and those that are stolen s m t + 1 s m t + 1 s_(mt+1)s_{m t+1}smt+1, and the aggregate size of products that are new to entering firms measured as the sum of those that are new products and those that are stolen s e t + 1 = 1 s c t + 1 s m t + 1 s e t + 1 = 1 s c t + 1 s m t + 1 s_(et+1)=1-s_(ct+1)-s_(mt+1)s_{e t+1}=1-s_{c t+1}-s_{m t+1}set+1=1sct+1smt+1. The average size of continuing products in continuing firms, new products in incumbent firms, and new products in entering firms is denoted by avsize c t + 1 = s c t + 1 f c t + 1 , a v s i z e m t + 1 = c t + 1 = s c t + 1 f c t + 1 , a v s i z e m t + 1 = _(ct+1)=(s_(ct+1))/(f_(ct+1)),avsize_(mt+1)={ }_{c t+1}=\frac{s_{c t+1}}{f_{c t+1}}, a v s i z e_{m t+1}=ct+1=sct+1fct+1,avsizemt+1= s m t + 1 f m t + 1 s m t + 1 f m t + 1 (s_(mt+1))/(f_(mt+1))\frac{s_{m t+1}}{f_{m t+1}}smt+1fmt+1 and avsize e t + 1 = s e t + 1 f e t + 1 e t + 1 = s e t + 1 f e t + 1 _(et+1)=(s_(et+1))/(f_(et+1))_{e t+1}=\frac{s_{e t+1}}{f_{e t+1}}et+1=set+1fet+1, respectively. Time averages of these variables are denoted with a bar.
We calibrate δ 0 , η e , η m δ 0 , η e , η m delta_(0),eta_(e),eta_(m)\delta_{0}, \eta_{e}, \eta_{m}δ0,ηe,ηm and the initial BGP values of h ( x ¯ m ) , ζ ( x ¯ c ) , x ¯ e h x ¯ m , ζ x ¯ c , x ¯ e h( bar(x)_(m)),zeta( bar(x)_(c)), bar(x)_(e)h\left(\bar{x}_{m}\right), \zeta\left(\bar{x}_{c}\right), \bar{x}_{e}h(x¯m),ζ(x¯c),x¯e to satisfy the following equations:
h ( x ¯ m ) = f ¯ m exp ( g ¯ M ) x ¯ e = f ¯ e exp ( g ¯ M ) ζ ( x ¯ c ) = avs i z e c exp ( ( ρ 1 ) g ¯ Z ) exp ( g ¯ M ) η m = avsize m exp ( ( ρ 1 ) g ¯ Z ) exp ( g ¯ M ) η e = avsize e exp ( ( ρ 1 ) g ¯ Z ) exp ( g ¯ M ) δ 0 = 1 δ m h ( x ¯ m ) δ e x ¯ e f ¯ c exp ( g ¯ M ) . h x ¯ m = f ¯ m exp g ¯ M x ¯ e = f ¯ e exp g ¯ M ζ x ¯ c = avs i z e ¯ c exp ( ρ 1 ) g ¯ Z exp g ¯ M η m = avsize m exp ( ρ 1 ) g ¯ Z exp g ¯ M η e = avsize e exp ( ρ 1 ) g ¯ Z exp g ¯ M δ 0 = 1 δ m h x ¯ m δ e x ¯ e f ¯ c exp g ¯ M . {:[h( bar(x)_(m))= bar(f)_(m)exp( bar(g)_(M))],[ bar(x)_(e)= bar(f)_(e)exp( bar(g)_(M))],[zeta( bar(x)_(c))=avs^(-) bar(ize)_(c)(exp((rho-1) bar(g)_(Z)))/(exp( bar(g)_(M)))],[eta_(m)=avsize_(m)(exp((rho-1) bar(g)_(Z)))/(exp( bar(g)_(M)))],[eta_(e)=avsize_(e)(exp((rho-1) bar(g)_(Z)))/(exp( bar(g)_(M)))],[delta_(0)=1-delta_(m)h( bar(x)_(m))-delta_(e) bar(x)_(e)- bar(f)_(c)exp( bar(g)_(M)).]:}\begin{aligned} & h\left(\bar{x}_{m}\right)=\bar{f}_{m} \exp \left(\bar{g}_{M}\right) \\ & \bar{x}_{e}=\bar{f}_{e} \exp \left(\bar{g}_{M}\right) \\ & \zeta\left(\bar{x}_{c}\right)=\operatorname{avs}^{-} \overline{i z e}_{c} \frac{\exp \left((\rho-1) \bar{g}_{Z}\right)}{\exp \left(\bar{g}_{M}\right)} \\ & \eta_{m}=\operatorname{avsize}_{m} \frac{\exp \left((\rho-1) \bar{g}_{Z}\right)}{\exp \left(\bar{g}_{M}\right)} \\ & \eta_{e}=\operatorname{avsize}_{e} \frac{\exp \left((\rho-1) \bar{g}_{Z}\right)}{\exp \left(\bar{g}_{M}\right)} \\ & \delta_{0}=1-\delta_{m} h\left(\bar{x}_{m}\right)-\delta_{e} \bar{x}_{e}-\bar{f}_{c} \exp \left(\bar{g}_{M}\right) . \end{aligned}h(x¯m)=f¯mexp(g¯M)x¯e=f¯eexp(g¯M)ζ(x¯c)=avsizecexp((ρ1)g¯Z)exp(g¯M)ηm=avsizemexp((ρ1)g¯Z)exp(g¯M)ηe=avsizeeexp((ρ1)g¯Z)exp(g¯M)δ0=1δmh(x¯m)δex¯ef¯cexp(g¯M).
Given initial policy ratios, ( 1 τ ¯ c 1 τ ¯ e ) 1 τ ¯ c 1 τ ¯ e ((1- bar(tau)_(c))/(1- bar(tau)_(e)))\left(\frac{1-\bar{\tau}_{c}}{1-\bar{\tau}_{e}}\right)(1τ¯c1τ¯e) and ( 1 τ ¯ c 1 τ ¯ m ) 1 τ ¯ c 1 τ ¯ m ((1- bar(tau)_(c))/(1- bar(tau)_(m)))\left(\frac{1-\bar{\tau}_{c}}{1-\bar{\tau}_{m}}\right)(1τ¯c1τ¯m), we calibrate ζ ( x ¯ c ) ζ x ¯ c zeta^(')( bar(x)_(c))\zeta^{\prime}\left(\bar{x}_{c}\right)ζ(x¯c) and h ( x ¯ m ) h x ¯ m h^(')( bar(x)_(m))h^{\prime}\left(\bar{x}_{m}\right)h(x¯m) using equations (64) and (65).
In order to calibrate the parameters h 1 , h 2 , ζ 1 h 1 , h 2 , ζ 1 h_(1),h_(2),zeta_(1)h_{1}, h_{2}, \zeta_{1}h1,h2,ζ1, and ζ 2 ζ 2 zeta_(2)\zeta_{2}ζ2, we must know the values of x ¯ c x ¯ c bar(x)_(c)\bar{x}_{c}x¯c and x ¯ m x ¯ m bar(x)_(m)\bar{x}_{m}x¯m. Note that our calibration procedure uses as an input a measure of ( x ¯ c + x ¯ m ) / Y ¯ r x ¯ c + x ¯ m / Y ¯ r ( bar(x)_(c)+ bar(x)_(m))// bar(Y)_(r)\left(\bar{x}_{c}+\bar{x}_{m}\right) / \bar{Y}_{r}(x¯c+x¯m)/Y¯r and implies a value of x ¯ e / Y ¯ r x ¯ e / Y ¯ r bar(x)_(e)// bar(Y)_(r)\bar{x}_{e} / \bar{Y}_{r}x¯e/Y¯r, but does not pin down x ¯ c x ¯ c bar(x)_(c)\bar{x}_{c}x¯c and x ¯ m x ¯ m bar(x)_(m)\bar{x}_{m}x¯m separately. To determine the value of x ¯ c x ¯ c bar(x)_(c)\bar{x}_{c}x¯c, we use the following logic. The contribution of investment in acquiring products each period to firm value must be nonnegative. That is, on a BGP, we must have v ¯ v ¯ bar(v)\bar{v}v¯ at least as large as the value that the firm would obtain if it were to set investment into acquiring new products equal to zero in every period. Given the assumption that
h ( 0 ) = 0 h ( 0 ) = 0 h(0)=0h(0)=0h(0)=0, this alternative value of incumbent firms on a BGP is given by
v ~ = [ 1 exp ( g ¯ Y ) 1 + R ¯ s ¯ c ] 1 [ ( 1 τ c o r p ) ( μ 1 μ ) ( 1 τ ¯ c ) P ¯ r t x ¯ c ( 1 + τ ¯ y ) Y ¯ t ] v ~ = 1 exp g ¯ Y 1 + R ¯ s ¯ c 1 1 τ c o r p μ 1 μ 1 τ ¯ c P ¯ r t x ¯ c 1 + τ ¯ y Y ¯ t tilde(v)=[1-(exp( bar(g)_(Y)))/(1+( bar(R))) bar(s)_(c)]^(-1)[(1-tau_(corp))((mu-1)/(mu))-(1- bar(tau)_(c))( bar(P)_(rt) bar(x)_(c))/((1+ bar(tau)_(y)) bar(Y)_(t))]\tilde{v}=\left[1-\frac{\exp \left(\bar{g}_{Y}\right)}{1+\bar{R}} \bar{s}_{c}\right]^{-1}\left[\left(1-\tau_{c o r p}\right)\left(\frac{\mu-1}{\mu}\right)-\left(1-\bar{\tau}_{c}\right) \frac{\bar{P}_{r t} \bar{x}_{c}}{\left(1+\bar{\tau}_{y}\right) \bar{Y}_{t}}\right]v~=[1exp(g¯Y)1+R¯s¯c]1[(1τcorp)(μ1μ)(1τ¯c)P¯rtx¯c(1+τ¯y)Y¯t]
If τ c = τ m τ c = τ m tau_(c)=tau_(m)\tau_{c}=\tau_{m}τc=τm, then
v ¯ = [ 1 exp ( g ¯ Y ) 1 + R ¯ ( s ¯ c + s ¯ m ) ] 1 [ ( 1 τ c o r p ) ( μ 1 μ ) ( 1 τ ¯ c ) P ¯ r t ( x ¯ c + x ¯ m ) ( 1 + τ ¯ y ) Y ¯ t ] v ¯ = 1 exp g ¯ Y 1 + R ¯ s ¯ c + s ¯ m 1 1 τ c o r p μ 1 μ 1 τ ¯ c P ¯ r t x ¯ c + x ¯ m 1 + τ ¯ y Y ¯ t bar(v)=[1-(exp( bar(g)_(Y)))/(1+( bar(R)))( bar(s)_(c)+ bar(s)_(m))]^(-1)[(1-tau_(corp))((mu-1)/(mu))-(1- bar(tau)_(c))( bar(P)_(rt)( bar(x)_(c)+ bar(x)_(m)))/((1+ bar(tau)_(y)) bar(Y)_(t))]\bar{v}=\left[1-\frac{\exp \left(\bar{g}_{Y}\right)}{1+\bar{R}}\left(\bar{s}_{c}+\bar{s}_{m}\right)\right]^{-1}\left[\left(1-\tau_{c o r p}\right)\left(\frac{\mu-1}{\mu}\right)-\left(1-\bar{\tau}_{c}\right) \frac{\bar{P}_{r t}\left(\bar{x}_{c}+\bar{x}_{m}\right)}{\left(1+\bar{\tau}_{y}\right) \bar{Y}_{t}}\right]v¯=[1exp(g¯Y)1+R¯(s¯c+s¯m)]1[(1τcorp)(μ1μ)(1τ¯c)P¯rt(x¯c+x¯m)(1+τ¯y)Y¯t]
The requirement that v ~ v ¯ v ~ v ¯ tilde(v) <= bar(v)\tilde{v} \leq \bar{v}v~v¯ implies that the research expenditures of incumbents on improving continuing products relative to value added must lie between the bounds
(89) P ¯ r t ( x ¯ c + x ¯ m ) ( 1 + τ ¯ y ) Y ¯ t P ¯ r t x ¯ c ( 1 + τ ¯ y ) Y ¯ t 1 exp ( g ¯ Y ) 1 + R ¯ s ¯ c 1 exp ( g ¯ Y ) 1 + R ¯ ( 1 s ¯ e ) P ¯ r t ( x ¯ c + x ¯ m ) ( 1 + τ ¯ y ) Y ¯ t exp ( g ¯ Y ) 1 + R ¯ s ¯ m 1 exp ( g ¯ Y ) 1 + R ¯ ( 1 s ¯ e ) ( 1 τ c o r p ) ( 1 τ ¯ c ) ( μ 1 μ ) (89) P ¯ r t x ¯ c + x ¯ m 1 + τ ¯ y Y ¯ t P ¯ r t x ¯ c 1 + τ ¯ y Y ¯ t 1 exp g ¯ Y 1 + R ¯ s ¯ c 1 exp ( g ¯ Y ) 1 + R ¯ 1 s ¯ e P ¯ r t x ¯ c + x ¯ m 1 + τ ¯ y Y ¯ t exp g ¯ Y 1 + R ¯ s ¯ m 1 exp g ¯ Y 1 + R ¯ 1 s ¯ e 1 τ c o r p 1 τ ¯ c μ 1 μ {:[(89)( bar(P)_(rt)( bar(x)_(c)+ bar(x)_(m)))/((1+ bar(tau)_(y)) bar(Y)_(t)) >= ( bar(P)_(rt) bar(x)_(c))/((1+ bar(tau)_(y)) bar(Y)_(t)) >= ],[(1-(exp( bar(g)_(Y)))/(1+( bar(R))) bar(s)_(c))/(1-(exp(( bar(g))Y))/(1+( bar(R)))(1- bar(s)_(e)))( bar(P)_(rt)( bar(x)_(c)+ bar(x)_(m)))/((1+ bar(tau)_(y)) bar(Y)_(t))-((exp( bar(g)_(Y)))/(1+( bar(R))) bar(s)_(m))/(1-(exp( bar(g)_(Y)))/(1+( bar(R)))(1- bar(s)_(e)))((1-tau_(corp)))/((1- bar(tau)_(c)))((mu-1)/(mu))]:}\begin{gather*} \frac{\bar{P}_{r t}\left(\bar{x}_{c}+\bar{x}_{m}\right)}{\left(1+\bar{\tau}_{y}\right) \bar{Y}_{t}} \geq \frac{\bar{P}_{r t} \bar{x}_{c}}{\left(1+\bar{\tau}_{y}\right) \bar{Y}_{t}} \geq \tag{89}\\ \frac{1-\frac{\exp \left(\bar{g}_{Y}\right)}{1+\bar{R}} \bar{s}_{c}}{1-\frac{\exp (\bar{g} Y)}{1+\bar{R}}\left(1-\bar{s}_{e}\right)} \frac{\bar{P}_{r t}\left(\bar{x}_{c}+\bar{x}_{m}\right)}{\left(1+\bar{\tau}_{y}\right) \bar{Y}_{t}}-\frac{\frac{\exp \left(\bar{g}_{Y}\right)}{1+\bar{R}} \bar{s}_{m}}{1-\frac{\exp \left(\bar{g}_{Y}\right)}{1+\bar{R}}\left(1-\bar{s}_{e}\right)} \frac{\left(1-\tau_{c o r p}\right)}{\left(1-\bar{\tau}_{c}\right)}\left(\frac{\mu-1}{\mu}\right) \end{gather*}(89)P¯rt(x¯c+x¯m)(1+τ¯y)Y¯tP¯rtx¯c(1+τ¯y)Y¯t1exp(g¯Y)1+R¯s¯c1exp(g¯Y)1+R¯(1s¯e)P¯rt(x¯c+x¯m)(1+τ¯y)Y¯texp(g¯Y)1+R¯s¯m1exp(g¯Y)1+R¯(1s¯e)(1τcorp)(1τ¯c)(μ1μ)
In our calibration, we set P ¯ r + x ¯ c ( 1 + τ ¯ y ) t ¯ t P ¯ r + x ¯ c 1 + τ ¯ y t ¯ t ( bar(P)_(r)+ bar(x)_(c))/((1+ bar(tau)_(y)) bar(t)_(t))\frac{\bar{P}_{r}+\bar{x}_{c}}{\left(1+\bar{\tau}_{y}\right) \bar{t}_{t}}P¯r+x¯c(1+τ¯y)t¯t in the middle point between the two bounds. Given values of x ¯ m , h ( x ¯ m ) x ¯ m , h x ¯ m bar(x)_(m),h( bar(x)_(m))\bar{x}_{m}, h\left(\bar{x}_{m}\right)x¯m,h(x¯m), and h ( x ¯ m ) h x ¯ m h^(')( bar(x)_(m))h^{\prime}\left(\bar{x}_{m}\right)h(x¯m), we determine the values of h 0 h 0 h_(0)h_{0}h0 and h 1 h 1 h_(1)h_{1}h1. Given values of x ¯ c , ζ ( x ¯ c ) x ¯ c , ζ x ¯ c bar(x)_(c),zeta( bar(x)_(c))\bar{x}_{c}, \zeta\left(\bar{x}_{c}\right)x¯c,ζ(x¯c), and ζ ( x ¯ c ) ζ x ¯ c zeta^(')( bar(x)_(c))\zeta^{\prime}\left(\bar{x}_{c}\right)ζ(x¯c) (which are assigned as described above, independently of ζ 2 ζ 2 zeta_(2)\zeta_{2}ζ2 ) and a value of 0 < ζ 2 < 1 0 < ζ 2 < 1 0 < zeta_(2) < 10<\zeta_{2}<10<ζ2<1, we determine the values of ζ 0 ζ 0 zeta_(0)\zeta_{0}ζ0 and ζ 1 ζ 1 zeta_(1)\zeta_{1}ζ1. For the policy exercises in which we do not set Θ Θ Theta\ThetaΘ at its upper bound of Θ e Θ e Theta_(e)\Theta_{e}Θe (which is independent of ζ 2 ζ 2 zeta_(2)\zeta_{2}ζ2 ), we set ζ 2 ζ 2 zeta_(2)\zeta_{2}ζ2 halfway between its two bounds, that is, ζ 2 = 0.5 ζ 2 = 0.5 zeta_(2)=0.5\zeta_{2}=0.5ζ2=0.5. This assumed value of ζ 2 ζ 2 zeta_(2)\zeta_{2}ζ2 implies that, with business stealing, Θ = 0.0093 < 0.0102 = Θ e Θ = 0.0093 < 0.0102 = Θ e Theta=0.0093 < 0.0102=Theta_(e)\Theta=0.0093<0.0102=\Theta_{e}Θ=0.0093<0.0102=Θe when considering proportional policy changes. 29 29 ^(29){ }^{29}29
We use LBD data to measure the necessary statistics to implement this calibration procedure. Specifically, we measure g ¯ M = 0.01 g ¯ M = 0.01 bar(g)_(M)=0.01\bar{g}_{M}=0.01g¯M=0.01 as the annual growth rate in the number of establishments (averaged between 1990 and 2014), f ¯ e = 0.0775 f ¯ e = 0.0775 bar(f)_(e)=0.0775\bar{f}_{e}=0.0775f¯e=0.0775 as the fraction of establishments in the data that are new and are owned by new firms, f ¯ m = 0.0227 f ¯ m = 0.0227 bar(f)_(m)=0.0227\bar{f}_{m}=0.0227f¯m=0.0227 as the fraction of establishments that are new and are owned by firms that are not new, and f ¯ c = 0.9 f ¯ c = 0.9 bar(f)_(c)=0.9\bar{f}_{c}=0.9f¯c=0.9 as the fraction of establishments that are not new. The employment shares of these three categories of establishments are s ¯ e = 0.027 , s ¯ m = 0.025 s ¯ e = 0.027 , s ¯ m = 0.025 bar(s)_(e)=0.027, bar(s)_(m)=0.025\bar{s}_{e}=0.027, \bar{s}_{m}=0.025s¯e=0.027,s¯m=0.025, and s ¯ c = 0.948 s ¯ c = 0.948 bar(s)_(c)=0.948\bar{s}_{c}=0.948s¯c=0.948. Our results are very similar if we calibrate the model using the statistics implied in Garcia-Macia et al. (2016).

Firm dynamics targets

Growth rate of measure of products, g ¯ M g ¯ M bar(g)_(M)\bar{g}_{M}g¯M
Fraction of products in entering firms, f ¯ e f ¯ e bar(f)_(e)\bar{f}_{e}f¯e
Fraction of new products in incumbent firms, f ¯ m f ¯ m bar(f)_(m)\bar{f}_{m}f¯m
Fraction of continuing products in incumbent firms, f ¯ c 0.900 f ¯ c 0.900 bar(f)_(c)quad0.900\bar{f}_{c} \quad 0.900f¯c0.900
Aggr. size of products in entering firms, s ¯ e s ¯ e bar(s)_(e)\bar{s}_{e}s¯e
Aggr. size of new products in incumbent firms, s ¯ m 0.025 s ¯ m 0.025 bar(s)_(m)quad0.025\bar{s}_{m} \quad 0.025s¯m0.025
Aggr. size of continuing products in incumbent firms, s ¯ c 0.948 s ¯ c 0.948 bar(s)_(c)quad0.948\bar{s}_{c} \quad 0.948s¯c0.948
Parameters
Exogenous exit rate, δ 0 δ 0 delta_(0)\delta_{0}δ0 0.091 or 0.071
Productivity step new products by entrants, η e η e eta_(e)\eta_{e}ηe 0.358
Productivity step new products by incumbents, η m η m eta_(m)\eta_{m}ηm 1.139
Innovation function new products by incumbents, h 0 h 0 h_(0)h_{0}h0 and h 1 h 1 h_(1)h_{1}h1 0.120 and 0.5
Innovation function products improvement by incumbents, ζ 0 , ζ 1 ζ 0 , ζ 1 zeta_(0),zeta_(1)\zeta_{0}, \zeta_{1}ζ0,ζ1 and ζ 2 ζ 2 zeta_(2)\zeta_{2}ζ2 0.995 , 0.268 0.995 , 0.268 0.995,0.2680.995,0.2680.995,0.268 and 0.5
Exogenous exit rate, delta_(0) 0.091 or 0.071 Productivity step new products by entrants, eta_(e) 0.358 Productivity step new products by incumbents, eta_(m) 1.139 Innovation function new products by incumbents, h_(0) and h_(1) 0.120 and 0.5 Innovation function products improvement by incumbents, zeta_(0),zeta_(1) and zeta_(2) 0.995,0.268 and 0.5| Exogenous exit rate, $\delta_{0}$ | 0.091 or 0.071 | | :--- | :--- | | Productivity step new products by entrants, $\eta_{e}$ | 0.358 | | Productivity step new products by incumbents, $\eta_{m}$ | 1.139 | | Innovation function new products by incumbents, $h_{0}$ and $h_{1}$ | 0.120 and 0.5 | | Innovation function products improvement by incumbents, $\zeta_{0}, \zeta_{1}$ and $\zeta_{2}$ | $0.995,0.268$ and 0.5 |
Other model outcomes
Lower bound P ¯ r t x ¯ c / ( ( 1 + τ ¯ y ) Y ¯ t ) P ¯ r t x ¯ c / 1 + τ ¯ y Y ¯ t bar(P)_(rt) bar(x)_(c)//((1+ bar(tau)_(y)) bar(Y)_(t))\bar{P}_{r t} \bar{x}_{c} /\left(\left(1+\bar{\tau}_{y}\right) \bar{Y}_{t}\right)P¯rtx¯c/((1+τ¯y)Y¯t) 0.032
Lower bound bar(P)_(rt) bar(x)_(c)//((1+ bar(tau)_(y)) bar(Y)_(t)) 0.032| Lower bound $\bar{P}_{r t} \bar{x}_{c} /\left(\left(1+\bar{\tau}_{y}\right) \bar{Y}_{t}\right)$ | 0.032 | | :--- | :--- |
P ¯ r t x ¯ c / ( ( 1 + τ ¯ y ) Y ¯ t ) 0.047 P ¯ r t x ¯ c / 1 + τ ¯ y Y ¯ t 0.047 bar(P)_(rt) bar(x)_(c)//((1+ bar(tau)_(y)) bar(Y)_(t))quad0.047\bar{P}_{r t} \bar{x}_{c} /\left(\left(1+\bar{\tau}_{y}\right) \bar{Y}_{t}\right) \quad 0.047P¯rtx¯c/((1+τ¯y)Y¯t)0.047
x ¯ m 0.036 x ¯ m 0.036 bar(x)_(m)quad0.036\bar{x}_{m} \quad 0.036x¯m0.036
x ¯ c 0.116 x ¯ c 0.116 bar(x)_(c)quad0.116\bar{x}_{c} \quad 0.116x¯c0.116
h ( x ¯ m ) 0.023 h x ¯ m 0.023 h( bar(x)_(m))quad0.023h\left(\bar{x}_{m}\right) \quad 0.023h(x¯m)0.023
h ( x ¯ m ) 0.314 h x ¯ m 0.314 h^(')( bar(x)_(m))quad0.314h^{\prime}\left(\bar{x}_{m}\right) \quad 0.314h(x¯m)0.314
ζ ( x ¯ c ) 1.087 ζ x ¯ c 1.087 zeta( bar(x)_(c))quad1.087\zeta\left(\bar{x}_{c}\right) \quad 1.087ζ(x¯c)1.087
ζ ( x ¯ c ) 0.393 ζ x ¯ c 0.393 zeta^(')( bar(x)_(c))quad0.393\zeta^{\prime}\left(\bar{x}_{c}\right) \quad 0.393ζ(x¯c)0.393
Table 3: Parameters and targets for non-linear transition dynamics

C. 4 Solving the model

Given parameter values assigned as described above, here we provide a summary of the steps to solve the model's BGP and transition dynamics assuming that the equilibrium allocations are interior (one needs to verify that the conjectured equilibrium that is obtained is interior). 30 30 ^(30){ }^{30}30
BGP The economy starts on an original BGP corresponding to policies τ ¯ c , τ ¯ m , τ ¯ e τ ¯ c , τ ¯ m , τ ¯ e bar(tau)_(c), bar(tau)_(m), bar(tau)_(e)\bar{\tau}_{c}, \bar{\tau}_{m}, \bar{\tau}_{e}τ¯c,τ¯m,τ¯e, and τ ¯ y τ ¯ y bar(tau)_(y)\bar{\tau}_{y}τ¯y. We normalize the level of scientific progress to be A r t = 1 A r t = 1 A_(rt)=1A_{r t}=1Art=1 at t = 0 t = 0 t=0t=0t=0. This gives A r t = A r t = A_(rt)=A_{r t}=Art= exp ( t g ¯ A r ) exp t g ¯ A r exp(t bar(g)_(Ar))\exp \left(t \bar{g}_{A r}\right)exp(tg¯Ar). We normalize the population to be L t = 1 L t = 1 L_(t)=1L_{t}=1Lt=1 at t = 0 t = 0 t=0t=0t=0. This gives L t = exp ( t g ¯ L ) L t = exp t g ¯ L L_(t)=exp(t bar(g)_(L))L_{t}=\exp \left(t \bar{g}_{L}\right)Lt=exp(tg¯L). We take as given parameters and observed growth rates of aggregate output and labor g ¯ Y g ¯ Y bar(g)_(Y)\bar{g}_{Y}g¯Y and g L g L g_(L)g_{L}gL.
Let l p t = L p t / L t l p t = L p t / L t l_(pt)=L_(pt)//L_(t)l_{p t}=L_{p t} / L_{t}lpt=Lpt/Lt and l r t = L r t / L t l r t = L r t / L t l_(rt)=L_(rt)//L_(t)l_{r t}=L_{r t} / L_{t}lrt=Lrt/Lt. These variables are constant at l ¯ p l ¯ p bar(l)_(p)\bar{l}_{p}l¯p and l ¯ r l ¯ r bar(l)_(r)\bar{l}_{r}l¯r on the initial BGP. Let k t = K t / exp ( t g ¯ Y ) k t = K t / exp t g ¯ Y k_(t)=K_(t)//exp(t bar(g)_(Y))k_{t}=K_{t} / \exp \left(t \bar{g}_{Y}\right)kt=Kt/exp(tg¯Y). This variable is constant at k ¯ k ¯ bar(k)\bar{k}k¯ on the initial BGP. Let v t = V t / ( ( 1 + τ ¯ y ) Y t ) v t = V t / 1 + τ ¯ y Y t v_(t)=V_(t)//((1+ bar(tau)_(y))Y_(t))v_{t}=V_{t} /\left(\left(1+\bar{\tau}_{y}\right) Y_{t}\right)vt=Vt/((1+τ¯y)Yt) and p r t = P r t / ( ( 1 + τ ¯ y ) Y t ) p r t = P r t / 1 + τ ¯ y Y t p_(rt)=P_(rt)//((1+ bar(tau)_(y))Y_(t))p_{r t}=P_{r t} /\left(\left(1+\bar{\tau}_{y}\right) Y_{t}\right)prt=Prt/((1+τ¯y)Yt). Both of these variables are constant on a BGP at v ¯ v ¯ bar(v)\bar{v}v¯ and p ¯ r p ¯ r bar(p)_(r)\bar{p}_{r}p¯r. Let z t = Z t / exp ( t g ¯ Z ) z t = Z t / exp t g ¯ Z z_(t)=Z_(t)//exp(t bar(g)_(Z))z_{t}=Z_{t} / \exp \left(t \bar{g}_{Z}\right)zt=Zt/exp(tg¯Z) where we solve for the BGP value g ¯ Z g ¯ Z bar(g)_(Z)\bar{g}_{Z}g¯Z below. Let z ¯ z ¯ bar(z)\bar{z}z¯ denote the BGP value of this variable. Let y t = Y t / exp ( t g ¯ Y ) y t = Y t / exp t g ¯ Y y_(t)=Y_(t)//exp(t bar(g)_(Y))y_{t}=Y_{t} / \exp \left(t \bar{g}_{Y}\right)yt=Yt/exp(tg¯Y) where we calibrate the BGP value g ¯ Y g ¯ Y bar(g)_(Y)\bar{g}_{Y}g¯Y. Let y ¯ y ¯ bar(y)\bar{y}y¯ denote the BGP value of this variable. Let c t = C t / exp ( t g ¯ Y ) c t = C t / exp t g ¯ Y c_(t)=C_(t)//exp(t bar(g)_(Y))c_{t}=C_{t} / \exp \left(t \bar{g}_{Y}\right)ct=Ct/exp(tg¯Y). Let c ¯ c ¯ bar(c)\bar{c}c¯ denote the BGP value of this variable. The BGP equations to be solved for the value of the state variables Z 0 Z 0 Z_(0)Z_{0}Z0 and K 0 K 0 K_(0)K_{0}K0 are as follows.
Growth rates g ¯ Z , g ¯ A r g ¯ Z , g ¯ A r bar(g)_(Z), bar(g)_(Ar)\bar{g}_{Z}, \bar{g}_{A r}g¯Z,g¯Ar :
g ¯ Z = ( 1 α ) ( g ¯ Y g ¯ L ) g ¯ A r = ( 1 ϕ ) g ¯ Z g ¯ L g ¯ Z = ( 1 α ) g ¯ Y g ¯ L g ¯ A r = ( 1 ϕ ) g ¯ Z g ¯ L {:[ bar(g)_(Z)=(1-alpha)( bar(g)_(Y)- bar(g)_(L))],[ bar(g)_(Ar)=(1-phi) bar(g)_(Z)- bar(g)_(L)]:}\begin{gathered} \bar{g}_{Z}=(1-\alpha)\left(\bar{g}_{Y}-\bar{g}_{L}\right) \\ \bar{g}_{A r}=(1-\phi) \bar{g}_{Z}-\bar{g}_{L} \end{gathered}g¯Z=(1α)(g¯Yg¯L)g¯Ar=(1ϕ)g¯Zg¯L
Interest rate and rental rate of physical capital R ¯ R ¯ bar(R)\bar{R}R¯ and R ¯ k R ¯ k bar(R)_(k)\bar{R}_{k}R¯k :
1 + R ¯ = exp ( γ ( g ¯ Y g ¯ L ) ) / β R ¯ = ( 1 τ c o r p ) ( R ¯ k d k ) 1 + R ¯ = exp γ g ¯ Y g ¯ L / β R ¯ = 1 τ c o r p R ¯ k d k {:[1+ bar(R)=exp(gamma( bar(g)_(Y)- bar(g)_(L)))//beta],[ bar(R)=(1-tau_(corp))( bar(R)_(k)-d_(k))]:}\begin{gathered} 1+\bar{R}=\exp \left(\gamma\left(\bar{g}_{Y}-\bar{g}_{L}\right)\right) / \beta \\ \bar{R}=\left(1-\tau_{c o r p}\right)\left(\bar{R}_{k}-d_{k}\right) \end{gathered}1+R¯=exp(γ(g¯Yg¯L))/βR¯=(1τcorp)(R¯kdk)
Innovative investment x ¯ c , x ¯ m , x ¯ e x ¯ c , x ¯ m , x ¯ e bar(x)_(c), bar(x)_(m), bar(x)_(e)\bar{x}_{c}, \bar{x}_{m}, \bar{x}_{e}x¯c,x¯m,x¯e, and Y ¯ r Y ¯ r bar(Y)_(r)\bar{Y}_{r}Y¯r :
1 τ ¯ c 1 τ ¯ e η e = ( 1 δ 0 δ m h ( x ¯ m ) δ e x ¯ e ) ζ ( x ¯ c ) 1 τ ¯ m 1 τ ¯ e η e = η m h ( x ¯ m ) g ¯ Z = 1 ρ 1 log ( ( 1 δ 0 δ m h ( x ¯ m ) δ e x ¯ e ) ζ ( x ¯ c ) + η m h ( x ¯ m ) + η e x ¯ e ) x ¯ c + x ¯ m + x ¯ e = Y ¯ r 1 τ ¯ c 1 τ ¯ e η e = 1 δ 0 δ m h x ¯ m δ e x ¯ e ζ x ¯ c 1 τ ¯ m 1 τ ¯ e η e = η m h x ¯ m g ¯ Z = 1 ρ 1 log 1 δ 0 δ m h x ¯ m δ e x ¯ e ζ x ¯ c + η m h x ¯ m + η e x ¯ e x ¯ c + x ¯ m + x ¯ e = Y ¯ r {:[(1- bar(tau)_(c))/(1- bar(tau)_(e))eta_(e)=(1-delta_(0)-delta_(m)h( bar(x)_(m))-delta_(e) bar(x)_(e))zeta^(')( bar(x)_(c))],[(1- bar(tau)_(m))/(1- bar(tau)_(e))eta_(e)=eta_(m)h^(')( bar(x)_(m))],[ bar(g)_(Z)=(1)/(rho-1)log((1-delta_(0)-delta_(m)h( bar(x)_(m))-delta_(e) bar(x)_(e))zeta( bar(x)_(c))+eta_(m)h( bar(x)_(m))+eta_(e) bar(x)_(e))],[ bar(x)_(c)+ bar(x)_(m)+ bar(x)_(e)= bar(Y)_(r)]:}\begin{gathered} \frac{1-\bar{\tau}_{c}}{1-\bar{\tau}_{e}} \eta_{e}=\left(1-\delta_{0}-\delta_{m} h\left(\bar{x}_{m}\right)-\delta_{e} \bar{x}_{e}\right) \zeta^{\prime}\left(\bar{x}_{c}\right) \\ \frac{1-\bar{\tau}_{m}}{1-\bar{\tau}_{e}} \eta_{e}=\eta_{m} h^{\prime}\left(\bar{x}_{m}\right) \\ \bar{g}_{Z}=\frac{1}{\rho-1} \log \left(\left(1-\delta_{0}-\delta_{m} h\left(\bar{x}_{m}\right)-\delta_{e} \bar{x}_{e}\right) \zeta\left(\bar{x}_{c}\right)+\eta_{m} h\left(\bar{x}_{m}\right)+\eta_{e} \bar{x}_{e}\right) \\ \bar{x}_{c}+\bar{x}_{m}+\bar{x}_{e}=\bar{Y}_{r} \end{gathered}1τ¯c1τ¯eηe=(1δ0δmh(x¯m)δex¯e)ζ(x¯c)1τ¯m1τ¯eηe=ηmh(x¯m)g¯Z=1ρ1log((1δ0δmh(x¯m)δex¯e)ζ(x¯c)+ηmh(x¯m)+ηex¯e)x¯c+x¯m+x¯e=Y¯r
Employment share of entering firms, value of intangible capital, price of the research good s ¯ e s ¯ e bar(s)_(e)\bar{s}_{e}s¯e, v ¯ v ¯ bar(v)\bar{v}v¯ and p ¯ r p ¯ r bar(p)_(r)\bar{p}_{r}p¯r :
s ¯ e = η e x ¯ e exp ( ( ρ 1 ) g ¯ Z ) (90) v ¯ = [ ( 1 τ c o r p ) μ 1 μ p ¯ r ( ( 1 τ ¯ c ) x ¯ c + ( 1 τ ¯ m ) x ¯ m ) ] 1 exp ( g ¯ Y ) 1 + R ¯ ( 1 s ¯ e ) ( 1 τ ¯ e ) p ¯ r x ¯ e = s ¯ e exp ( g ¯ Y ) 1 + R ¯ v ¯ s ¯ e = η e x ¯ e exp ( ρ 1 ) g ¯ Z (90) v ¯ = 1 τ c o r p μ 1 μ p ¯ r 1 τ ¯ c x ¯ c + 1 τ ¯ m x ¯ m 1 exp g ¯ Y 1 + R ¯ 1 s ¯ e 1 τ ¯ e p ¯ r x ¯ e = s ¯ e exp g ¯ Y 1 + R ¯ v ¯ {:[ bar(s)_(e)=(eta_(e) bar(x)_(e))/(exp((rho-1) bar(g)_(Z)))],[(90) bar(v)=([(1-tau_(corp))(mu-1)/(mu)- bar(p)_(r)((1- bar(tau)_(c)) bar(x)_(c)+(1- bar(tau)_(m)) bar(x)_(m))])/(1-(exp( bar(g)_(Y)))/(1+( bar(R)))(1- bar(s)_(e)))],[(1- bar(tau)_(e)) bar(p)_(r) bar(x)_(e)= bar(s)_(e)(exp( bar(g)_(Y)))/(1+( bar(R))) bar(v)]:}\begin{gather*} \bar{s}_{e}=\frac{\eta_{e} \bar{x}_{e}}{\exp \left((\rho-1) \bar{g}_{Z}\right)} \\ \bar{v}=\frac{\left[\left(1-\tau_{c o r p}\right) \frac{\mu-1}{\mu}-\bar{p}_{r}\left(\left(1-\bar{\tau}_{c}\right) \bar{x}_{c}+\left(1-\bar{\tau}_{m}\right) \bar{x}_{m}\right)\right]}{1-\frac{\exp \left(\bar{g}_{Y}\right)}{1+\bar{R}}\left(1-\bar{s}_{e}\right)} \tag{90}\\ \left(1-\bar{\tau}_{e}\right) \bar{p}_{r} \bar{x}_{e}=\bar{s}_{e} \frac{\exp \left(\bar{g}_{Y}\right)}{1+\bar{R}} \bar{v} \end{gather*}s¯e=ηex¯eexp((ρ1)g¯Z)(90)v¯=[(1τcorp)μ1μp¯r((1τ¯c)x¯c+(1τ¯m)x¯m)]1exp(g¯Y)1+R¯(1s¯e)(1τ¯e)p¯rx¯e=s¯eexp(g¯Y)1+R¯v¯
Note that by combining the last two expressions, we can rewrite v ¯ v ¯ bar(v)\bar{v}v¯ as
(91) v ¯ = [ ( 1 τ c o r p ) μ 1 μ p ¯ r ( ( 1 τ ¯ c ) x ¯ c + ( 1 τ ¯ m ) x ¯ m + ( 1 τ ¯ e ) x ¯ e ) ] 1 exp ( g ¯ Y ) 1 + R ¯ (91) v ¯ = 1 τ c o r p μ 1 μ p ¯ r 1 τ ¯ c x ¯ c + 1 τ ¯ m x ¯ m + 1 τ ¯ e x ¯ e 1 exp g ¯ Y 1 + R ¯ {:(91) bar(v)=([(1-tau_(corp))(mu-1)/(mu)- bar(p)_(r)((1- bar(tau)_(c)) bar(x)_(c)+(1- bar(tau)_(m)) bar(x)_(m)+(1- bar(tau)_(e)) bar(x)_(e))])/(1-(exp( bar(g)_(Y)))/(1+( bar(R)))):}\begin{equation*} \bar{v}=\frac{\left[\left(1-\tau_{c o r p}\right) \frac{\mu-1}{\mu}-\bar{p}_{r}\left(\left(1-\bar{\tau}_{c}\right) \bar{x}_{c}+\left(1-\bar{\tau}_{m}\right) \bar{x}_{m}+\left(1-\bar{\tau}_{e}\right) \bar{x}_{e}\right)\right]}{1-\frac{\exp \left(\bar{g}_{Y}\right)}{1+\bar{R}}} \tag{91} \end{equation*}(91)v¯=[(1τcorp)μ1μp¯r((1τ¯c)x¯c+(1τ¯m)x¯m+(1τ¯e)x¯e)]1exp(g¯Y)1+R¯
Innovation intensity and allocation of labor i ¯ r , l ¯ p , l ¯ r i ¯ r , l ¯ p , l ¯ r bar(i)_(r), bar(l)_(p), bar(l)_(r)\bar{i}_{r}, \bar{l}_{p}, \bar{l}_{r}i¯r,l¯p,l¯r :
i ¯ r = p ¯ r Y ¯ r l ¯ p l ¯ r = 1 α μ 1 i ¯ r l ¯ p + l ¯ r = 1 i ¯ r = p ¯ r Y ¯ r l ¯ p l ¯ r = 1 α μ 1 i ¯ r l ¯ p + l ¯ r = 1 {:[ bar(i)_(r)= bar(p)_(r) bar(Y)_(r)],[( bar(l)_(p))/( bar(l)_(r))=(1-alpha)/(mu)(1)/( bar(i)_(r))],[ bar(l)_(p)+ bar(l)_(r)=1]:}\begin{gathered} \bar{i}_{r}=\bar{p}_{r} \bar{Y}_{r} \\ \frac{\bar{l}_{p}}{\bar{l}_{r}}=\frac{1-\alpha}{\mu} \frac{1}{\bar{i}_{r}} \\ \bar{l}_{p}+\bar{l}_{r}=1 \end{gathered}i¯r=p¯rY¯rl¯pl¯r=1αμ1i¯rl¯p+l¯r=1
Levels of aggregate productivity z ¯ z ¯ bar(z)\bar{z}z¯ :
Y ¯ r = z ¯ ϕ 1 l ¯ r Y ¯ r = z ¯ ϕ 1 l ¯ r bar(Y)_(r)= bar(z)^(phi-1) bar(l)_(r)\bar{Y}_{r}=\bar{z}^{\phi-1} \bar{l}_{r}Y¯r=z¯ϕ1l¯r
Output, consumption, and capital stock y ¯ , c ¯ y ¯ , c ¯ bar(y), bar(c)\bar{y}, \bar{c}y¯,c¯ and k ¯ k ¯ bar(k)\bar{k}k¯ :
R ¯ k = ( 1 + τ y ) α μ y ¯ k ¯ y ¯ = z ¯ α l ¯ p 1 α c ¯ = y ¯ ( exp ( g ¯ Y ) + ( 1 d k ) ) k ¯ R ¯ k = 1 + τ y α μ y ¯ k ¯ y ¯ = z ¯ α l ¯ p 1 α c ¯ = y ¯ exp g ¯ Y + 1 d k k ¯ {:[ bar(R)_(k)=(1+tau_(y))(alpha )/(mu)(( bar(y)))/(( bar(k)))],[ bar(y)= bar(z)^(alpha) bar(l)_(p)^(1-alpha)],[ bar(c)= bar(y)-(exp( bar(g)_(Y))+(1-d_(k))) bar(k)]:}\begin{gathered} \bar{R}_{k}=\left(1+\tau_{y}\right) \frac{\alpha}{\mu} \frac{\bar{y}}{\bar{k}} \\ \bar{y}=\bar{z}^{\alpha} \bar{l}_{p}^{1-\alpha} \\ \bar{c}=\bar{y}-\left(\exp \left(\bar{g}_{Y}\right)+\left(1-d_{k}\right)\right) \bar{k} \end{gathered}R¯k=(1+τy)αμy¯k¯y¯=z¯αl¯p1αc¯=y¯(exp(g¯Y)+(1dk))k¯
New BGP We consider a permanent change in innovation policies from τ ¯ c , τ ¯ m , τ ¯ e τ ¯ c , τ ¯ m , τ ¯ e bar(tau)_(c), bar(tau)_(m), bar(tau)_(e)\bar{\tau}_{c}, \bar{\tau}_{m}, \bar{\tau}_{e}τ¯c,τ¯m,τ¯e to τ ¯ c , τ ¯ m , τ ¯ e τ ¯ c , τ ¯ m , τ ¯ e bar(tau)_(c)^('), bar(tau)_(m)^('), bar(tau)_(e)^(')\bar{\tau}_{c}^{\prime}, \bar{\tau}_{m}^{\prime}, \bar{\tau}_{e}^{\prime}τ¯c,τ¯m,τ¯e that is unanticipated and starts in period t = 0 t = 0 t=0t=0t=0. Thus, the economy starts in period t = 0 t = 0 t=0t=0t=0 with the state variables A r 0 = L 0 = 1 A r 0 = L 0 = 1 A_(r0)=L_(0)=1A_{r 0}=L_{0}=1Ar0=L0=1, and Z ¯ 0 , K ¯ 0 Z ¯ 0 , K ¯ 0 bar(Z)_(0), bar(K)_(0)\bar{Z}_{0}, \bar{K}_{0}Z¯0,K¯0 solved for above. The economy follows a transition path to a new BGP. We can use the same procedure as above to compute the new BGP, which we denote with primes. The terminal condition we will use when we solve for the transition dynamics is that we end up at the new BGP allocation.
Transition dynamics The two states of the model are z t z t z_(t)z_{t}zt and k t k t k_(t)k_{t}kt. Given z 0 , k 0 z 0 , k 0 z_(0),k_(0)z_{0}, k_{0}z0,k0, we wish to solve for a sequence { z t + 1 , k t + 1 } 0 T z t + 1 , k t + 1 0 T {z_(t+1),k_(t+1)}_(0)^(T)\left\{z_{t+1}, k_{t+1}\right\}_{0}^{T}{zt+1,kt+1}0T, as well as the remaining intra-period variables that satisfy the Euler equations and the terminal conditions that z t , k t z t , k t z_(t),k_(t)z_{t}, k_{t}zt,kt converges to the new BGP z ¯ , k ¯ z ¯ , k ¯ bar(z)^('), bar(k)^(')\bar{z}^{\prime}, \bar{k}^{\prime}z¯,k¯.
Let Y t Y t Y_(t)Y_{t}Yt denote the vector of intra-period variables:
Y t = ( x c t , x m t , x e t , Y r t , l r t , l p t , i r t , p r t , y t , c t ) Y t = x c t , x m t , x e t , Y r t , l r t , l p t , i r t , p r t , y t , c t Y_(t)=(x_(ct),x_(mt),x_(et),Y_(rt),l_(rt),l_(pt),i_(rt),p_(rt),y_(t),c_(t))Y_{t}=\left(x_{c t}, x_{m t}, x_{e t}, Y_{r t}, l_{r t}, l_{p t}, i_{r t}, p_{r t}, y_{t}, c_{t}\right)Yt=(xct,xmt,xet,Yrt,lrt,lpt,irt,prt,yt,ct)
We first show how we solve for Y t Y t Y_(t)Y_{t}Yt, given { z t , z t + 1 , k t , k t + 1 } z t , z t + 1 , k t , k t + 1 {z_(t),z_(t+1),k_(t),k_(t+1)}\left\{z_{t}, z_{t+1}, k_{t}, k_{t+1}\right\}{zt,zt+1,kt,kt+1}. Specifically, we solve for ( x c t , x m t , x e t , Y r t , l r t , l p t , i r t , p r t ) x c t , x m t , x e t , Y r t , l r t , l p t , i r t , p r t (x_(ct),x_(mt),x_(et),Y_(rt),l_(rt),l_(pt),i_(rt),p_(rt))\left(x_{c t}, x_{m t}, x_{e t}, Y_{r t}, l_{r t}, l_{p t}, i_{r t}, p_{r t}\right)(xct,xmt,xet,Yrt,lrt,lpt,irt,prt) given ( z t , z t + 1 ) z t , z t + 1 (z_(t),z_(t+1))\left(z_{t}, z_{t+1}\right)(zt,zt+1). We then solve for ( y t , c t ) y t , c t (y_(t),c_(t))\left(y_{t}, c_{t}\right)(yt,ct) given ( k t , k t + 1 ) k t , k t + 1 (k_(t),k_(t+1))\left(k_{t}, k_{t+1}\right)(kt,kt+1).
Allocation of innovative investment x c t , x m t , x e t , Y r t x c t , x m t , x e t , Y r t x_(ct),x_(mt),x_(et),Y_(rt)x_{c t}, x_{m t}, x_{e t}, Y_{r t}xct,xmt,xet,Yrt (assuming that the allocation is interior):
(92) log z t + 1 log z t + g ¯ z = 1 ρ 1 log ( ( 1 δ 0 δ m h ( x m t ) δ e x e t ) ζ ( x c t ) + η m h ( x m t ) + η e x e t ) (93) 1 τ ¯ c 1 τ ¯ e η e = ( 1 δ 0 δ m h ( x m t ) δ e x e t ) ) ζ ( x c t ) (94) 1 τ ¯ m 1 τ ¯ e η e = η m h ( x m t ) (92) log z t + 1 log z t + g ¯ z = 1 ρ 1 log 1 δ 0 δ m h x m t δ e x e t ζ x c t + η m h x m t + η e x e t (93) 1 τ ¯ c 1 τ ¯ e η e = 1 δ 0 δ m h x m t δ e x e t ζ x c t (94) 1 τ ¯ m 1 τ ¯ e η e = η m h x m t {:[(92)log z_(t+1)-log z_(t)+ bar(g)_(z)=(1)/(rho-1)log((1-delta_(0)-delta_(m)h(x_(mt))-delta_(e)x_(et))zeta(x_(ct))+eta_(m)h(x_(mt))+eta_(e)x_(et))],[(93){:(1- bar(tau)_(c))/(1- bar(tau)_(e))eta_(e)=(1-delta_(0)-delta_(m)h(x_(mt))-delta_(e)x_(et)))zeta^(')(x_(ct))],[(94)(1- bar(tau)_(m))/(1- bar(tau)_(e))eta_(e)=eta_(m)h^(')(x_(mt))]:}\begin{gather*} \log z_{t+1}-\log z_{t}+\bar{g}_{z}=\frac{1}{\rho-1} \log \left(\left(1-\delta_{0}-\delta_{m} h\left(x_{m t}\right)-\delta_{e} x_{e t}\right) \zeta\left(x_{c t}\right)+\eta_{m} h\left(x_{m t}\right)+\eta_{e} x_{e t}\right) \tag{92}\\ \left.\frac{1-\bar{\tau}_{c}}{1-\bar{\tau}_{e}} \eta_{e}=\left(1-\delta_{0}-\delta_{m} h\left(x_{m t}\right)-\delta_{e} x_{e t}\right)\right) \zeta^{\prime}\left(x_{c t}\right) \tag{93}\\ \frac{1-\bar{\tau}_{m}}{1-\bar{\tau}_{e}} \eta_{e}=\eta_{m} h^{\prime}\left(x_{m t}\right) \tag{94} \end{gather*}(92)logzt+1logzt+g¯z=1ρ1log((1δ0δmh(xmt)δexet)ζ(xct)+ηmh(xmt)+ηexet)(93)1τ¯c1τ¯eηe=(1δ0δmh(xmt)δexet))ζ(xct)(94)1τ¯m1τ¯eηe=ηmh(xmt)
(95) x c t + x m t + x e t = Y r t (95) x c t + x m t + x e t = Y r t {:(95)x_(ct)+x_(mt)+x_(et)=Y_(rt):}\begin{equation*} x_{c t}+x_{m t}+x_{e t}=Y_{r t} \tag{95} \end{equation*}(95)xct+xmt+xet=Yrt
Allocation of labor l r t , l p t l r t , l p t l_(rt),l_(pt)l_{r t}, l_{p t}lrt,lpt :
(96) Y r t = z t ϕ 1 l r t (97) l p t + l r t = 1 (96) Y r t = z t ϕ 1 l r t (97) l p t + l r t = 1 {:[(96)Y_(rt)=z_(t)^(phi-1)l_(rt)],[(97)l_(pt)+l_(rt)=1]:}\begin{gather*} Y_{r t}=z_{t}^{\phi-1} l_{r t} \tag{96}\\ l_{p t}+l_{r t}=1 \tag{97} \end{gather*}(96)Yrt=ztϕ1lrt(97)lpt+lrt=1
Price of the research and innovation intensity i r t , p r t : i r t , p r t : i_(rt),p_(rt):i_{r t}, p_{r t}:irt,prt:
(98) l p t l r t = 1 α μ 1 i r t (99) i r t = p r t Y r t (98) l p t l r t = 1 α μ 1 i r t (99) i r t = p r t Y r t {:[(98)(l_(pt))/(l_(rt))=(1-alpha)/(mu)(1)/(i_(rt))],[(99)i_(rt)=p_(rt)Y_(rt)]:}\begin{gather*} \frac{l_{p t}}{l_{r t}}=\frac{1-\alpha}{\mu} \frac{1}{i_{r t}} \tag{98}\\ i_{r t}=p_{r t} Y_{r t} \tag{99} \end{gather*}(98)lptlrt=1αμ1irt(99)irt=prtYrt
Output and consumption y t , c t y t , c t y_(t),c_(t)y_{t}, c_{t}yt,ct :
(100) y t = z t k t α l p t 1 α (101) c t = y t + ( 1 d k ) k t exp ( g ¯ Y ) k t + 1 (100) y t = z t k t α l p t 1 α (101) c t = y t + 1 d k k t exp g ¯ Y k t + 1 {:[(100)y_(t)=z_(t)k_(t)^(alpha)l_(pt)^(1-alpha)],[(101)c_(t)=y_(t)+(1-d_(k))k_(t)-exp( bar(g)_(Y))k_(t+1)]:}\begin{gather*} y_{t}=z_{t} k_{t}^{\alpha} l_{p t}^{1-\alpha} \tag{100}\\ c_{t}=y_{t}+\left(1-d_{k}\right) k_{t}-\exp \left(\bar{g}_{Y}\right) k_{t+1} \tag{101} \end{gather*}(100)yt=ztktαlpt1α(101)ct=yt+(1dk)ktexp(g¯Y)kt+1
We now consider the Euler equations that must be satisfied.
Standard physical capital Euler equation:
(102) R t = ( 1 τ corp ) ( ( 1 + τ y ) α μ y t + 1 k t + 1 d k ) (103) 1 = ( 1 + R t ) β ~ exp ( g ¯ Y ) ( c t + 1 c t ) γ (102) R t = 1 τ corp  1 + τ y α μ y t + 1 k t + 1 d k (103) 1 = 1 + R t β ~ exp g ¯ Y c t + 1 c t γ {:[(102)R_(t)=(1-tau_("corp "))((1+tau_(y))(alpha )/(mu)(y_(t+1))/(k_(t+1))-d_(k))],[(103)1=(1+R_(t)) tilde(beta)exp(- bar(g)_(Y))((c_(t+1))/(c_(t)))^(-gamma)]:}\begin{align*} R_{t} & =\left(1-\tau_{\text {corp }}\right)\left(\left(1+\tau_{y}\right) \frac{\alpha}{\mu} \frac{y_{t+1}}{k_{t+1}}-d_{k}\right) \tag{102}\\ 1 & =\left(1+R_{t}\right) \tilde{\beta} \exp \left(-\bar{g}_{Y}\right)\left(\frac{c_{t+1}}{c_{t}}\right)^{-\gamma} \tag{103} \end{align*}(102)Rt=(1τcorp )((1+τy)αμyt+1kt+1dk)(103)1=(1+Rt)β~exp(g¯Y)(ct+1ct)γ
Value function and zero profit at entry condition:
v t = ( 1 τ c o r p ) μ 1 μ p r t ( ( 1 τ c ) x c t + ( 1 τ m ) x m t ) + exp ( g ¯ Y ( ρ 1 ) g ¯ Z ) 1 + R t v t + 1 ( y t + 1 y t ) ( z t z t + 1 ) ρ 1 [ ( 1 δ 0 δ m h ( x m t ) δ e x e t ) ) ζ ( x c t ) + η m h ( x m t ) ] and v t = 1 τ c o r p μ 1 μ p r t 1 τ c x c t + 1 τ m x m t + exp g ¯ Y ( ρ 1 ) g ¯ Z 1 + R t v t + 1 y t + 1 y t z t z t + 1 ρ 1 1 δ 0 δ m h x m t δ e x e t ζ x c t + η m h x m t  and  {:[v_(t)=(1-tau_(corp))(mu-1)/(mu)-p_(rt)((1-tau_(c))x_(ct)+(1-tau_(m))x_(mt))+],[{:(exp( bar(g)_(Y)-(rho-1) bar(g)_(Z)))/(1+R_(t))v_(t+1)((y_(t+1))/(y_(t)))((z_(t))/(z_(t+1)))^(rho-1)[(1-delta_(0)-delta_(m)h(x_(mt))-delta_(e)x_(et)))zeta(x_(ct))+eta_(m)h(x_(mt))]],[" and "]:}\begin{aligned} & v_{t}=\left(1-\tau_{c o r p}\right) \frac{\mu-1}{\mu}-p_{r t}\left(\left(1-\tau_{c}\right) x_{c t}+\left(1-\tau_{m}\right) x_{m t}\right)+ \\ & \left.\frac{\exp \left(\bar{g}_{Y}-(\rho-1) \bar{g}_{Z}\right)}{1+R_{t}} v_{t+1}\left(\frac{y_{t+1}}{y_{t}}\right)\left(\frac{z_{t}}{z_{t+1}}\right)^{\rho-1}\left[\left(1-\delta_{0}-\delta_{m} h\left(x_{m t}\right)-\delta_{e} x_{e t}\right)\right) \zeta\left(x_{c t}\right)+\eta_{m} h\left(x_{m t}\right)\right] \\ & \text { and } \end{aligned}vt=(1τcorp)μ1μprt((1τc)xct+(1τm)xmt)+exp(g¯Y(ρ1)g¯Z)1+Rtvt+1(yt+1yt)(ztzt+1)ρ1[(1δ0δmh(xmt)δexet))ζ(xct)+ηmh(xmt)] and 
(105) ( 1 τ e t ) p r t = η e exp ( g ¯ Y ( ρ 1 ) g ¯ Z ) 1 + R t v t + 1 ( y t + 1 y t ) ( z t z t + 1 ) ρ 1 (105) 1 τ e t p r t = η e exp g ¯ Y ( ρ 1 ) g ¯ Z 1 + R t v t + 1 y t + 1 y t z t z t + 1 ρ 1 {:(105)(1-tau_(et))p_(rt)=eta_(e)(exp( bar(g)_(Y)-(rho-1) bar(g)_(Z)))/(1+R_(t))v_(t+1)((y_(t+1))/(y_(t)))((z_(t))/(z_(t+1)))^(rho-1):}\begin{equation*} \left(1-\tau_{e t}\right) p_{r t}=\eta_{e} \frac{\exp \left(\bar{g}_{Y}-(\rho-1) \bar{g}_{Z}\right)}{1+R_{t}} v_{t+1}\left(\frac{y_{t+1}}{y_{t}}\right)\left(\frac{z_{t}}{z_{t+1}}\right)^{\rho-1} \tag{105} \end{equation*}(105)(1τet)prt=ηeexp(g¯Y(ρ1)g¯Z)1+Rtvt+1(yt+1yt)(ztzt+1)ρ1
Let
X t = ( z t , k t , v t 1 ) X t = z t , k t , v t 1 X_(t)=(z_(t),k_(t),v_(t-1))X_{t}=\left(z_{t}, k_{t}, v_{t-1}\right)Xt=(zt,kt,vt1)
denote the vector of state variables extended with the lagged value of v t v t v_(t)v_{t}vt. If we specify the triple X t , X t + 1 , X t + 2 X t , X t + 1 , X t + 2 X_(t),X_(t+1),X_(t+2)X_{t}, X_{t+1}, X_{t+2}Xt,Xt+1,Xt+2, we can solve for Y t Y t Y_(t)Y_{t}Yt and Y t + 1 Y t + 1 Y_(t+1)Y_{t+1}Yt+1, and we calculate the Euler equations above. We then solve for the path of { X t , Y t } X t , Y t {X_(t),Y_(t)}\left\{X_{t}, Y_{t}\right\}{Xt,Yt} using standard linear or nonlinear methods.
Solving transition taking as given a path of innovation intensities or research labor We now consider the problem of finding innovation policies to implement an equilibrium with a pre-specified path of the allocation of labor to research { l r t } l r t {l_(rt)}\left\{l_{r t}\right\}{lrt} or, equivalently given equation (98), a path for the innovation intensity of economy { i r t } i r t {i_(rt)}\left\{i_{r t}\right\}{irt}. We begin with an analysis of our simple model in which only entering firms invest in innovation. We then extend the analysis to include investment by incumbent firms. We impose that the sequence { l r t } l r t {l_(rt)}\left\{l_{r t}\right\}{lrt} converges to a constant value l ¯ r l ¯ r bar(l)_(r)^(')\bar{l}_{r}^{\prime}l¯r associated with the new BGP.
Given z 0 , k 0 z 0 , k 0 z_(0),k_(0)z_{0}, k_{0}z0,k0, consider a given path of the allocation of research labor { l r t } l r t {l_(rt)}\left\{l_{r t}\right\}{lrt} (or, similarly, the innovation intensity of the economy { i r t } ) i r t {:{i_(rt)})\left.\left\{i_{r t}\right\}\right){irt}). Assume that a given level of the tax τ c o r p τ c o r p tau_(corp)\tau_{c o r p}τcorp is fixed. We solve for the implied allocation and the sequence of entry subsidies that implement this allocation as an equilibrium as follows.
Equations (92), (95), (96), (97), (98), and (99) are used recursively to construct the implied sequences in the transition for { z t + 1 , x e t , Y r t , l p t , i r t , p r t } z t + 1 , x e t , Y r t , l p t , i r t , p r t {z_(t+1),x_(et),Y_(rt),l_(pt),i_(rt),p_(rt)}\left\{z_{t+1}, x_{e t}, Y_{r t}, l_{p t}, i_{r t}, p_{r t}\right\}{zt+1,xet,Yrt,lpt,irt,prt}. The equilibrium sequences of consumption, output, physical capital and the real interest rate c t , y t , k t + 1 , R t c t , y t , k t + 1 , R t c_(t),y_(t),k_(t+1),R_(t)c_{t}, y_{t}, k_{t+1}, R_{t}ct,yt,kt+1,Rt in the transition are given as the solutions to equations (100), (101), (102), and (103) together with the terminal condition that the capital stock converge to its new BGP value lim t k t + 1 = k ¯ lim t k t + 1 = k ¯ lim_(t rarr oo)k_(t+1)= bar(k)^(')\lim _{t \rightarrow \infty} k_{t+1}=\bar{k}^{\prime}limtkt+1=k¯. These equations are those equations characterizing equilibrium in the growth model with time-varying exogenous supply of production labor l p t l p t l_(pt)l_{p t}lpt and aggregate productivity z t z t z_(t)z_{t}zt.
Solving equation (104) recursively, the value of a product of size one relative to output v t v t v_(t)v_{t}vt is given by
v t = [ d t + k = 1 d t + k ( y t + k y t ) ( z t z t + k ) ρ 1 exp ( g ¯ Y ( ρ 1 ) g ¯ Z ) k j = 0 k 1 1 δ c t + j 1 + R t + j ] v t = d t + k = 1 d t + k y t + k y t z t z t + k ρ 1 exp g ¯ Y ( ρ 1 ) g ¯ Z k j = 0 k 1 1 δ c t + j 1 + R t + j v_(t)=[d_(t)+sum_(k=1)^(oo)d_(t+k)((y_(t+k))/(y_(t)))((z_(t))/(z_(t+k)))^(rho-1)exp (( bar(g))Y-(rho-1) bar(g)_(Z))^(k)prod_(j=0)^(k-1)(1-delta_(ct+j))/(1+R_(t+j))]v_{t}=\left[d_{t}+\sum_{k=1}^{\infty} d_{t+k}\left(\frac{y_{t+k}}{y_{t}}\right)\left(\frac{z_{t}}{z_{t+k}}\right)^{\rho-1} \exp \left(\bar{g} Y-(\rho-1) \bar{g}_{Z}\right)^{k} \prod_{j=0}^{k-1} \frac{1-\delta_{c t+j}}{1+R_{t+j}}\right]vt=[dt+k=1dt+k(yt+kyt)(ztzt+k)ρ1exp(g¯Y(ρ1)g¯Z)kj=0k11δct+j1+Rt+j]
where 1 δ c t = 1 δ 0 δ e x e t 1 δ c t = 1 δ 0 δ e x e t 1-delta_(ct)=1-delta_(0)-delta_(e)x_(et)1-\delta_{c t}=1-\delta_{0}-\delta_{e} x_{e t}1δct=1δ0δexet and d t = ( 1 τ c o r p ) μ 1 μ d t = 1 τ c o r p μ 1 μ d_(t)=(1-tau_(corp))(mu-1)/(mu)d_{t}=\left(1-\tau_{c o r p}\right) \frac{\mu-1}{\mu}dt=(1τcorp)μ1μ. The sequence of innovation policies { τ e t } τ e t {tau_(et)}\left\{\tau_{e t}\right\}{τet} that implement this allocation as an equilibrium is then given from equation (105).
Observe that when we extend this result to the model with innovative investment
by incumbents, we can no longer uniquely identify the feasible allocation corresponding to a given path of the allocation of labor to research { l r t } l r t {l_(rt)}\left\{l_{r t}\right\}{lrt}. Instead, we must specify a rule for interior investment by incumbents x c t x c t x_(ct)x_{c t}xct and x m t x m t x_(mt)x_{m t}xmt as a function of Y r t Y r t Y_(rt)Y_{r t}Yrt that allows for x e t x e t x_(et)x_{e t}xet strictly positive. One such rule would be to specify that x c t / Y r t x c t / Y r t x_(ct)//Y_(rt)x_{c t} / Y_{r t}xct/Yrt and x m t / Y r t x m t / Y r t x_(mt)//Y_(rt)x_{m t} / Y_{r t}xmt/Yrt are constant fractions that sum to less than one. Then, from equation (95), we can solve for an interior value of x e t x e t x_(et)x_{e t}xet. With such a rule for allocating innovative investment across firms, we can repeat the procedure for finding innovation policies that implement the implied allocation as an equilibrium as follows.
Again equations (92), (95), (96), (97), (98), and (99) and our rule for allocating innovative investment by incumbents are used recursively to construct the implied sequences in the transition for { z t + 1 , x c t , x m t , x e t , Y r t , l p t , i r t , p r t } z t + 1 , x c t , x m t , x e t , Y r t , l p t , i r t , p r t {z_(t+1),x_(ct),x_(mt),x_(et),Y_(rt),l_(pt),i_(rt),p_(rt)}\left\{z_{t+1}, x_{c t}, x_{m t}, x_{e t}, Y_{r t}, l_{p t}, i_{r t}, p_{r t}\right\}{zt+1,xct,xmt,xet,Yrt,lpt,irt,prt}. The equilibrium sequences of consumption, output, physical capital, and the real interest rate c t , y t , k t + 1 , R t c t , y t , k t + 1 , R t c_(t),y_(t),k_(t+1),R_(t)c_{t}, y_{t}, k_{t+1}, R_{t}ct,yt,kt+1,Rt in the transition are given as the solutions to equations (100), (101), (102), and (103) together with the terminal condition that the capital stock converge to its new BGP value lim t k t + 1 = k ¯ lim t k t + 1 = k ¯ lim_(t rarr oo)k_(t+1)= bar(k)^(')\lim _{t \rightarrow \infty} k_{t+1}=\bar{k}^{\prime}limtkt+1=k¯.
The problem of finding policies to implement this allocation as an equilibrium is somewhat more complex than is the case in the simple model. We can use equations (93) and (94) to find the required values of ( 1 τ c t ) / ( 1 τ e t ) 1 τ c t / 1 τ e t (1-tau_(ct))//(1-tau_(et))\left(1-\tau_{c t}\right) /\left(1-\tau_{e t}\right)(1τct)/(1τet) and ( 1 τ m t ) / ( 1 τ e t ) 1 τ m t / 1 τ e t (1-tau_(mt))//(1-tau_(et))\left(1-\tau_{m t}\right) /\left(1-\tau_{e t}\right)(1τmt)/(1τet) by plugging the allocation into the right-hand side of these equations. We then have to solve for the entire sequence of entry subsidies { τ e t } τ e t {tau_(et)}\left\{\tau_{e t}\right\}{τet} as follows. The dividend to an incumbent product is now
d t = ( 1 τ c o r p ) μ 1 μ ( 1 τ e t ) p r t ( 1 τ c t 1 τ e t x c t + 1 τ m t 1 τ e t x m t ) d t = 1 τ c o r p μ 1 μ 1 τ e t p r t 1 τ c t 1 τ e t x c t + 1 τ m t 1 τ e t x m t d_(t)=(1-tau_(corp))(mu-1)/(mu)-(1-tau_(et))p_(rt)((1-tau_(ct))/(1-tau_(et))x_(ct)+(1-tau_(mt))/(1-tau_(et))x_(mt))d_{t}=\left(1-\tau_{c o r p}\right) \frac{\mu-1}{\mu}-\left(1-\tau_{e t}\right) p_{r t}\left(\frac{1-\tau_{c t}}{1-\tau_{e t}} x_{c t}+\frac{1-\tau_{m t}}{1-\tau_{e t}} x_{m t}\right)dt=(1τcorp)μ1μ(1τet)prt(1τct1τetxct+1τmt1τetxmt)
The value of a product of size one relative to output v t v t v_(t)v_{t}vt is given as above, using this expression for dividends and with 1 δ c t 1 δ c t 1-delta_(ct)1-\delta_{c t}1δct now including business stealing by incumbent firms. The sequence of entry subsidies that implements this allocation as an equilibrium is then given by the solution to equation (105).
Alternatively to specifying interior paths for x c t / Y r t x c t / Y r t x_(ct)//Y_(rt)x_{c t} / Y_{r t}xct/Yrt and x m t / Y r t x m t / Y r t x_(mt)//Y_(rt)x_{m t} / Y_{r t}xmt/Yrt, we can specify timevarying policy ratios 1 τ c t 1 τ e t 1 τ c t 1 τ e t (1-tau_(ct))/(1-tau_(et))\frac{1-\tau_{c t}}{1-\tau_{e t}}1τct1τet and 1 τ m t 1 τ e t 1 τ m t 1 τ e t (1-tau_(mt))/(1-tau_(et))\frac{1-\tau_{m t}}{1-\tau_{e t}}1τmt1τet, and solve for x c t , x m t x c t , x m t x_(ct),x_(mt)x_{c t}, x_{m t}xct,xmt, and x e t x e t x_(et)x_{e t}xet using equations (95), (93), and (94). At this point, we must verify that the solution to this system of equations implies allocations of innovative investment that are interior. After solving for a timevarying path of { τ e t } τ e t {tau_(et)}\left\{\tau_{e t}\right\}{τet} as described above, we calculate { τ c t , τ m t } τ c t , τ m t {tau_(ct),tau_(mt)}\left\{\tau_{c t}, \tau_{m t}\right\}{τct,τmt} given the innovation policy ratios.

C. 5 Additional numerical results

In this section, we report additional details and figures for our innovation policy counterfactuals considered in Section 7. We first consider our baseline proportional policy
changes. We then consider nonproportional policy changes. Finally, we consider proportional policy changes that are accompanied by annual exogenous productivity shocks.
Figures 1 and 2 display, for each of our four specifications, the transition dynamics (solving the model fully nonlinearly) of aggregate productivity and output in the first 100 and 20 years, respectively, after the proportional policy change that increases research labor permanently by 10 % 10 % 10%10 \%10%. Figure 3 displays the 20-year transition dynamics of fiscal expenditures E / Y E / Y E//YE / YE/Y and the innovation subsidy rate τ e t τ e t tau_(et)\tau_{e t}τet that produces the 10 % 10 % 10%10 \%10% permanent increase in research labor. Recall that, with uniform innovation policies, the fiscal cost of these policies in terms of the change in E / Y E / Y E//YE / YE/Y from the initial BGP to the new BGP required to induce a given change in the innovation intensity of the economy is given as in Proposition 4. In Figure 3 we see that the increase in innovation subsidies in the early phase of the transition is smaller than in the new BGP, especially with low intertemporal knowledge spillovers ϕ ϕ phi\phiϕ. With low ϕ ϕ phi\phiϕ, the price of the research good is expected to rise quickly during the transition, so that (due to intertemporal substitution) innovation subsidies do not have to be as large in order to induce the same increase in total expenditures.
Figure 4 compares the path of aggregate productivity and output calculated using the log-linear approximation (as in Table 1) and the nonlinear solution method. The small differences between the linear and nonlinear solutions (which vanish in the new BGP) result in very small differences in welfare. For example, with high ϕ ϕ phi\phiϕ and no business stealing (in which the differences between the solutions in the first 100 years are the largest in Figure 4), the consumption equivalent changes in welfare are very similar: 20.96 % 20.96 % 20.96%20.96 \%20.96% when solving the model nonlinearly and 20.08% when solving the model via linear approximation.
We next consider nonproportional changes in innovation subsidies, which satisfy restriction (32). Specifically, we solve for the dynamics of innovation subsidies for entrants, τ e t τ e t tau_(et)\tau_{e t}τet, such that research labor increases by 10 % 10 % 10%10 \%10% permanently and both 1 τ c t 1 τ e t 1 τ c t 1 τ e t (1-tau_(ct))/(1-tau_(et))\frac{1-\tau_{c t}}{1-\tau_{e t}}1τct1τet and 1 τ m t 1 τ e t 1 τ m t 1 τ e t (1-tau_(mt))/(1-tau_(et))\frac{1-\tau_{m t}}{1-\tau_{e t}}1τmt1τet fall from 1 to 0.95 . That is, innovation investments by incumbents are disproportionately subsidized relative to innovation investments by entrants.
We first assume that there is no business stealing. Figure 5 displays the transition dynamics of aggregate productivity and output in the first 100 years when solving the model via linearization (around the initial BGP) and nonlinearly. Given that the initial BGP allocation of innovative investments is conditionally efficient, Proposition 5 applies. Up to a first-order approximation, the transition dynamics of aggregate productivity and output only depend on the aggregate change in research labor and are the same for proportional and nonproportional policy changes (so that the transition dynamics coincide with those displayed in Figure 4 under the linearized solution). Figure 5 shows that the difference between the linearized (using the expressions referenced in Proposition 5) and nonlinear
transition dynamics are quite small. With ϕ = 1.6 ϕ = 1.6 phi=-1.6\phi=-1.6ϕ=1.6, the resulting change in welfare is 2.6 % 2.6 % 2.6%2.6 \%2.6% based on the linearized solution and 2.5 % 2.5 % 2.5%2.5 \%2.5% based on the nonlinearized solution. With ϕ = 0.96 ϕ = 0.96 phi=0.96\phi=0.96ϕ=0.96, the resulting change in welfare is 20.1 % 20.1 % 20.1%20.1 \%20.1% based on the linearized solution and 20.6 % 20.6 % 20.6%20.6 \%20.6% based on the nonlinearized solution.
We next assume that there is business stealing. In this case, the initial BGP allocation of innovative investment is not conditionally efficient, so Proposition 5 does not apply. Since policy changes are nonproportional, Proposition 7 does not apply either. However, in this case we can apply Proposition 6, which provides a first-order approximation around the new BGP. Figure 6 displays the transition dynamics of aggregate productivity and output in the first 100 years, in the presence of business stealing, when solving the model via linearization (as indicated in Proposition 6) and nonlinearly. In this case, there are additional output gains from reallocating innovative investment from entrants to incumbent firms (since Θ c Θ c Theta_(c)\Theta_{c}Θc and Θ m Θ m Theta_(m)\Theta_{m}Θm are larger than Θ e Θ e Theta_(e)\Theta_{e}Θe ). The welfare gains when solving the model nonlinearly are 5.4 % 5.4 % 5.4%5.4 \%5.4% with low ϕ ϕ phi\phiϕ and 16.2 % 16.2 % 16.2%16.2 \%16.2% with high ϕ ϕ phi\phiϕ (versus 1.7 % 1.7 % 1.7%1.7 \%1.7% and 6.9 % 6.9 % 6.9%6.9 \%6.9%, respectively, when considering a proportional policy change). The differences between the linearized and nonlinear transition dynamics are quite modest given the large policy change that we consider. With ϕ = 1.6 ϕ = 1.6 phi=-1.6\phi=-1.6ϕ=1.6, the resulting change in welfare is 5.2 % 5.2 % 5.2%5.2 \%5.2% based on the linearized solution and 5.4 % 5.4 % 5.4%5.4 \%5.4% based on the nonlinearized solution. With ϕ = 0.96 ϕ = 0.96 phi=0.96\phi=0.96ϕ=0.96, the resulting change in welfare is 14.7 % 14.7 % 14.7%14.7 \%14.7% based on the linearized solution and 16.2 % 16.2 % 16.2%16.2 \%16.2% based on the nonlinearized solution.
Our results in Section 7 suggest that it would be hard to verify whether innovation policies yield large output and welfare gains using medium-term data on the response of aggregates to changes in innovation policies. We illustrate this point in Figure 7. In that figure, we show results obtained from simulating the response of aggregates in our model (with and without business stealing, with low and high ϕ ϕ phi\phiϕ ) to our baseline proportional increase in innovation subsidies in an extended version of our model with Hicks-neutral AR1 productivity shocks with a persistence of 0.9 and an annual standard deviation of 0.018. We introduce these shocks as a proxy for business cycle shocks around the BGP. We show histograms generated from 3,000 simulations of the log-linearized model for the first 20 years of the transition. The units on the horizontal axis show the log of the ratio of detrended output (in the upper panels) and productivity (in the lower panels) at the end of the 20th year of transition to initial output or productivity. The vertical axis shows the frequency of the corresponding outcome for output or productivity. The red bars show results for the model with low intertemporal knowledge spillovers ϕ ϕ phi\phiϕ, and the blue bars show the results with high spillovers ϕ ϕ phi\phiϕ. We can observe in each panel that the distribution represented by the blue bars is slightly to the right of that represented by
the red bars. But it is also clear in each panel that, using either output or productivity, it would be very hard to distinguish the degree of intertemporal knowledge spillovers (and, hence, the long-term effects from this innovation subsidy) in aggregate time series data even if we had the benefit of a true policy experiment.
Figure 1: 100-year transition dynamics to a 10% permanent increase in research labor via proportional innovation policies, nonlinear solution
Figure 2: 20-year transition dynamics to a 10% permanent increase in research labor via proportional innovation policies, nonlinear solution
Figure 3: Fiscal expenditures and innovation subsidy rate, 20-year transition dynamics to a 10 % 10 % 10%10 \%10% permanent increase in research labor via proportional innovation policies, nonlinear solution
Figure 4: 100-year transition dynamics to a 10% permanent increase in research labor via proportional innovation policies, nonlinear versus linear solution
Figure 5: 100-year transition dynamics to a 10% permanent increase in research labor via nonproportional innovation policies, no business stealing, nonlinear versus linear solution
Figure 6: 100-year transition dynamics to a 10 % 10 % 10%10 \%10% permanent increase in research labor via nonproportional innovation policies, with business stealing, nonlinear versus linear solution
Figure 7: Histogram of 20-year increase in aggregate output and productivity to a permanent 10 % 10 % 10%10 \%10% increase in research labor, including productivity shocks

D Discussion of models not nested in our framework

As discussed in Section 4.4, our model nests several influential models in the literature on firms' innovative investments and aggregate growth. In constructing our model, we rely on three key assumptions in deriving our analytical results: (i) the markup μ μ mu\muμ is constant across products and time, (ii) the costs and benefits of innovative investments by incumbent firms scale with firm size, and (iii) all incumbent firms share the same technologies for innovative investment. Some recent papers in the literature examine models of firms' investments in innovation in which one or more of these key assumptions do not hold. We discuss some of these alternative model specifications here and the extent to which our results may be applied to these models.
Consider first our assumption that markups μ μ mu\muμ are constant across intermediate goods
and time. We rely on this assumption to derive the formula (6) for aggregate productivity that plays a central role in our derivation of our analytical results. If we consider an alternative model in which both productivity indices z z zzz and markups μ μ mu\muμ on intermediate goods had support on a grid, and if we let M t ( z n , μ j ) M t z n , μ j M_(t)(z_(n),mu_(j))M_{t}\left(z_{n}, \mu_{j}\right)Mt(zn,μj) denote the measure of intermediate goods with productivity index z n z n z_(n)z_{n}zn and markup μ j μ j mu_(j)\mu_{j}μj, then aggregate productivity Z t Z t Z_(t)Z_{t}Zt would be given by
Z t = ( n j z n ρ 1 μ j 1 ρ M t ( z n , μ j ) ) ρ / ( ρ 1 ) n j z n ρ 1 μ j ρ M t ( z n , μ j ) Z t = n j z n ρ 1 μ j 1 ρ M t z n , μ j ρ / ( ρ 1 ) n j z n ρ 1 μ j ρ M t z n , μ j Z_(t)=((sum_(n)sum_(j)z_(n)^(rho-1)mu_(j)^(1-rho)M_(t)(z_(n),mu_(j)))^(rho//(rho-1)))/(sum_(n)sum_(j)z_(n)^(rho-1)mu_(j)^(-rho)M_(t)(z_(n),mu_(j)))Z_{t}=\frac{\left(\sum_{n} \sum_{j} z_{n}^{\rho-1} \mu_{j}^{1-\rho} M_{t}\left(z_{n}, \mu_{j}\right)\right)^{\rho /(\rho-1)}}{\sum_{n} \sum_{j} z_{n}^{\rho-1} \mu_{j}^{-\rho} M_{t}\left(z_{n}, \mu_{j}\right)}Zt=(njznρ1μj1ρMt(zn,μj))ρ/(ρ1)njznρ1μjρMt(zn,μj)
In general, then, in a model in which markups vary across intermediate goods, one must keep track of the evolution of the joint distribution of productivity indices z z zzz and markups μ μ mu\muμ across products to compute the evolution of aggregate productivity. 31 31 ^(31){ }^{31}31
Peters (2016) presents a model that emphasizes the interaction of firms' investments in innovation and their markups. He introduces a Neo-Schumpeterian model in which incumbent firms invest to improve their own products, and entering and incumbent firms invest to acquire new products. An incumbent firm that innovates on its own product charges a higher markup on its product relative to an entering firm or an incumbent firm acquiring a new product. As a result of this assumption, the joint distribution of markups and productivity indices across products varies over time. Hence, there is no simple analog to the equation (28) we derive in Lemma 3 linking aggregate innovative investments by firms and aggregate productivity growth. Instead, one must keep track of the evolution of the measure M t ( z , μ ) M t ( z , μ ) M_(t)(z,mu)M_{t}(z, \mu)Mt(z,μ) to compute the dynamics of aggregate productivity implied by his model. Innovation policy interacts with competition policy in that investments in innovation by entrants and incumbent firms seeking to acquire new products have an additional impact on the growth of aggregate productivity that arises from the impact of this type of innovation on the portion of products sold at a low markup.
Consider next our assumption that the costs and benefits of innovative investments scale with firm size. This assumption is key to our derivation of equation (28) in Lemma 3 linking aggregate innovative investments by firms and aggregate productivity growth. Akcigit and Kerr (2018) estimate an alternative model of firms' investments in innovation and firm dynamics in which investments by incumbent firms in acquiring new products do not scale proportionally with firm size. In general, in a model of this kind, one must
31 31 ^(31){ }^{31}31 If the distribution of markups across products is independent of the productivity index of the product and if the marginal distribution of markups is constant over time, then the terms j μ j 1 ρ M t ( z n , μ j ) j μ j 1 ρ M t z n , μ j sum_(j)mu_(j)^(1-rho)M_(t)(z_(n),mu_(j))\sum_{j} \mu_{j}^{1-\rho} M_{t}\left(z_{n}, \mu_{j}\right)jμj1ρMt(zn,μj) and j μ j ρ M t ( z n , μ j ) j μ j ρ M t z n , μ j sum_(j)mu_(j)^(-rho)M_(t)(z_(n),mu_(j))\sum_{j} \mu_{j}^{-\rho} M_{t}\left(z_{n}, \mu_{j}\right)jμjρMt(zn,μj) do not vary with z n z n z_(n)z_{n}zn or with t t ttt. In this case, the formula above for aggregate productivity reduces to our formula (6) times a constant. In this case, we can extend our analytical results to cover a model with markups that vary across intermediate goods.
keep track of the distribution of incumbent firms by size to compute the transition dynamics of aggregate productivity implied by the model. In Appendix E.4, we analyze a version of our model in which investments to acquire new products do not scale up one to one with firm size, but instead scale up with the number of products that the firm has (this specification of our model is nested in the generalized model of Akcigit and Kerr 2018). We discuss the extent to which we can extend our analytical results and measurement procedure to this alternative model.
Finally, consider our assumption that all incumbent firms share the same technologies for innovative investment. Lentz and Mortensen (2008), Lentz and Mortensen (2016), and Luttmer (2011) present examples of a Neo-Schumpeterian model (in the first two papers) and an expanding varieties model (in the third paper) in which different types of incumbent firms have different technologies for innovating. To nest these models in our framework, we would index the parameter η m η m eta_(m)\eta_{m}ηm and the functions h ( ) h ( ) h(*)h(\cdot)h() and ζ ( ) ζ ( ) zeta(*)\zeta(\cdot)ζ() characterizing the investment technologies for incumbent firms to acquire new products and to improve their own products respectively by a firm "type" i i iii. The evolution of aggregate productivity in these models is then a function of the measure of intermediate goods produced by firms of each type together with the aggregate investments x m i x m i x_(m)^(i)x_{m}^{i}xmi and x c i x c i x_(c)^(i)x_{c}^{i}xci of incumbent firms of each type. To compute the transition dynamics of aggregate productivity implied by these models, one would have to keep track of the evolution of the distribution of intermediate goods produced by each type of firm.

E Variations of baseline model

E. 1 Occupation choice

Suppose that workers each period draw a productivity a a aaa to work in the research sector, where a a aaa is drawn from a cumulative distribution function F ( a ) F ( a ) F(a)F(a)F(a) that is Pareto with minimum 1 and slope coefficient χ > 1 χ > 1 chi > 1\chi>1χ>1. There are two wages, W p t W p t W_(pt)W_{p t}Wpt and W r t W r t W_(rt)W_{r t}Wrt. For the marginal agent,
a ¯ t W r t = W p t a ¯ t W r t = W p t bar(a)_(t)W_(rt)=W_(pt)\bar{a}_{t} W_{r t}=W_{p t}a¯tWrt=Wpt
Given that the minimum value of a a aaa is 1 , any interior equilibrium with positive production requires W r t W p t W r t W p t W_(rt) <= W_(pt)W_{r t} \leq W_{p t}WrtWpt. The aggregate supplies of production and research labor (relative to the total labor force, which grows exogenously) are
l p t = F ( a ¯ t ) = 1 a ¯ t χ l p t = F a ¯ t = 1 a ¯ t χ l_(pt)=F( bar(a)_(t))=1- bar(a)_(t)^(-chi)l_{p t}=F\left(\bar{a}_{t}\right)=1-\bar{a}_{t}^{-\chi}lpt=F(a¯t)=1a¯tχ
l r t = a ¯ t a f ( a ) d a = χ χ 1 a ¯ t 1 χ l r t = a ¯ t a f ( a ) d a = χ χ 1 a ¯ t 1 χ l_(rt)=int_( bar(a)_(t))^(oo)af(a)da=(chi)/(chi-1) bar(a)_(t)^(1-chi)l_{r t}=\int_{\bar{a}_{t}}^{\infty} a f(a) d a=\frac{\chi}{\chi-1} \bar{a}_{t}^{1-\chi}lrt=a¯taf(a)da=χχ1a¯t1χ
The ratio of labor l r t / l p t l r t / l p t l_(rt)//l_(pt)l_{r t} / l_{p t}lrt/lpt and the ratio of wages W r t / W p t W r t / W p t W_(rt)//W_(pt)W_{r t} / W_{p t}Wrt/Wpt are determined by
W r t l r t W p t l p t = μ ( 1 α ) i r t l r t l p t = χ χ 1 ( W p t W r t ) 1 χ 1 ( W p t W r t ) χ W r t l r t W p t l p t = μ ( 1 α ) i r t l r t l p t = χ χ 1 W p t W r t 1 χ 1 W p t W r t χ {:[(W_(rt)l_(rt))/(W_(pt)l_(pt))=(mu)/((1-alpha))i_(rt)],[(l_(rt))/(l_(pt))=(chi)/(chi-1)(((W_(pt))/(W_(rt)))^(1-chi))/(1-((W_(pt))/(W_(rt)))^(-chi))]:}\begin{gathered} \frac{W_{r t} l_{r t}}{W_{p t} l_{p t}}=\frac{\mu}{(1-\alpha)} i_{r t} \\ \frac{l_{r t}}{l_{p t}}=\frac{\chi}{\chi-1} \frac{\left(\frac{W_{p t}}{W_{r t}}\right)^{1-\chi}}{1-\left(\frac{W_{p t}}{W_{r t}}\right)^{-\chi}} \end{gathered}WrtlrtWptlpt=μ(1α)irtlrtlpt=χχ1(WptWrt)1χ1(WptWrt)χ
These two equations replace equation (13) in our baseline model to solve for l r t / l p t l r t / l p t l_(rt)//l_(pt)l_{r t} / l_{p t}lrt/lpt as a function of i r t i r t i_(rt)i_{r t}irt. Note that as χ χ chi\chiχ goes to infinity, W r t / W p t W r t / W p t W_(rt)//W_(pt)W_{r t} / W_{p t}Wrt/Wpt must converge to 1 in order for l r t / l p t l r t / l p t l_(rt)//l_(pt)l_{r t} / l_{p t}lrt/lpt to be finite. The elasticity of l r t l r t l_(rt)l_{r t}lrt with respect to i r t i r t i_(rt)i_{r t}irt is
( log l r t log l ¯ r ) = ( χ 1 ) ( χ 1 ) ( 1 + W ¯ r l ¯ r W ¯ p l ¯ p ) + 1 ( log i r t log i ¯ r ) log l r t log l ¯ r = ( χ 1 ) ( χ 1 ) 1 + W ¯ r l ¯ r W ¯ p l ¯ p + 1 log i r t log i ¯ r (log l_(rt)^(')-log bar(l)_(r))=((chi-1))/((chi-1)(1+( bar(W)_(r) bar(l)_(r))/( bar(W)_(p) bar(l)_(p)))+1)(log i_(rt)^(')-log bar(i)_(r))\left(\log l_{r t}^{\prime}-\log \bar{l}_{r}\right)=\frac{(\chi-1)}{(\chi-1)\left(1+\frac{\bar{W}_{r} \bar{l}_{r}}{\bar{W}_{p} \bar{l}_{p}}\right)+1}\left(\log i_{r t}^{\prime}-\log \bar{i}_{r}\right)(loglrtlogl¯r)=(χ1)(χ1)(1+W¯rl¯rW¯pl¯p)+1(logirtlogi¯r)
When χ χ chi\chiχ converges to 1 (high worker heterogeneity), the elasticity of l r t l r t l_(rt)l_{r t}lrt with respect to i r t i r t i_(rt)i_{r t}irt converges to 0 . When χ χ chi\chiχ converges to infinity (no worker heterogeneity), this elasticity converges to l ¯ p l ¯ p bar(l)_(p)\bar{l}_{p}l¯p, as in our baseline model.

E. 2 Goods and labor used as inputs in research

We consider an extension in which research production uses both labor and the consumption good, as in the lab-equipment model of Rivera-Batiz and Romer (1991), and discuss the central changes to our analytic results. Specifically, the production of the research good is given by
(106) Y r t = A r t Z t ϕ 1 L r t λ X t 1 λ (106) Y r t = A r t Z t ϕ 1 L r t λ X t 1 λ {:(106)Y_(rt)=A_(rt)Z_(t)^(phi-1)L_(rt)^(lambda)X_(t)^(1-lambda):}\begin{equation*} Y_{r t}=A_{r t} Z_{t}^{\phi-1} L_{r t}^{\lambda} X_{t}^{1-\lambda} \tag{106} \end{equation*}(106)Yrt=ArtZtϕ1LrtλXt1λ
and the resource constraint of the final consumption good is
C t + K t + 1 ( 1 d k ) K t + X t = Y t C t + K t + 1 1 d k K t + X t = Y t C_(t)+K_(t+1)-(1-d_(k))K_(t)+X_(t)=Y_(t)C_{t}+K_{t+1}-\left(1-d_{k}\right) K_{t}+X_{t}=Y_{t}Ct+Kt+1(1dk)Kt+Xt=Yt
Given this production technology, the BGP growth rate of aggregate productivity is given by g ¯ Z = δ ¯ A r + & L 1 ϕ ~ g ¯ Z = δ ¯ A r + & ¯ L 1 ϕ ~ bar(g)_(Z)=( bar(delta)_(A_(r))+ bar(&)_(L))/(1-( tilde(phi)))\bar{g}_{Z}=\frac{\bar{\delta}_{A_{r}}+\overline{\&}_{L}}{1-\tilde{\phi}}g¯Z=δ¯Ar+&L1ϕ~, where ϕ ~ = ϕ + 1 λ 1 α ϕ ~ = ϕ + 1 λ 1 α tilde(phi)=phi+(1-lambda)/(1-alpha)\tilde{\phi}=\phi+\frac{1-\lambda}{1-\alpha}ϕ~=ϕ+1λ1α. The condition for semi-endogenous growth is ϕ ~ < 1 ϕ ~ < 1 tilde(phi) < 1\tilde{\phi}<1ϕ~<1 (equality for endogenous growth).
Revenues from the production of the research good are divided as follows:
(107) W t L r t = λ P r t Y r t , and X t = ( 1 λ ) P r t Y r t (107) W t L r t = λ P r t Y r t ,  and  X t = ( 1 λ ) P r t Y r t {:(107)W_(t)L_(rt)=lambdaP_(rt)Y_(rt)","" and "X_(t)=(1-lambda)P_(rt)Y_(rt):}\begin{equation*} W_{t} L_{r t}=\lambda P_{r t} Y_{r t}, \text { and } X_{t}=(1-\lambda) P_{r t} Y_{r t} \tag{107} \end{equation*}(107)WtLrt=λPrtYrt, and Xt=(1λ)PrtYrt
The analog to equation (13), relating the allocation of labor between production and research to the innovation intensity, is
(108) l r t = i r t i r t + ( 1 α ) μ λ and l p t = 1 l r t (108) l r t = i r t i r t + ( 1 α ) μ λ  and  l p t = 1 l r t {:(108)l_(rt)=(i_(rt))/(i_(rt)+((1-alpha))/(mu lambda))" and "l_(pt)=1-l_(rt):}\begin{equation*} l_{r t}=\frac{i_{r t}}{i_{r t}+\frac{(1-\alpha)}{\mu \lambda}} \text { and } l_{p t}=1-l_{r t} \tag{108} \end{equation*}(108)lrt=irtirt+(1α)μλ and lpt=1lrt
where i r t P r t Y r t / ( ( 1 + τ y ) Y t ) i r t P r t Y r t / 1 + τ y Y t i_(rt)-=P_(rt)Y_(rt)//((1+tau_(y))Y_(t))i_{r t} \equiv P_{r t} Y_{r t} /\left(\left(1+\tau_{y}\right) Y_{t}\right)irtPrtYrt/((1+τy)Yt). Our analytical results need to be modified for two reasons. First, for a given share of production labor in output ( 1 α ) / μ ( 1 α ) / μ (1-alpha)//mu(1-\alpha) / \mu(1α)/μ, the elasticity of research labor l r t l r t l_(rt)l_{r t}lrt with respect to the innovation intensity of the economy i r t i r t i_(rt)i_{r t}irt is decreasing in λ λ lambda\lambdaλ. A higher share of goods in production of the research good (lower λ λ lambda\lambdaλ ) increases the sensitivity of l r t l r t l_(rt)l_{r t}lrt with respect to i r t i r t i_(rt)i_{r t}irt.
Second, by equations (5), (106), (107), and R k t = ( 1 + τ y ) α μ Y t K t R k t = 1 + τ y α μ Y t K t R_(kt)=(1+tau_(y))(alpha )/(mu)(Y_(t))/(K_(t))R_{k t}=\left(1+\tau_{y}\right) \frac{\alpha}{\mu} \frac{Y_{t}}{K_{t}}Rkt=(1+τy)αμYtKt, we have
Y r t = κ A r t Z t Φ ~ 1 ( K t Y t ) α ( 1 λ ) 1 α L r t Y r t = κ A r t Z t Φ ~ 1 K t Y t α ( 1 λ ) 1 α L r t Y_(rt)=kappaA_(rt)Z_(t)^( tilde(Phi)-1)((K_(t))/(Y_(t)))^((alpha(1-lambda))/(1-alpha))L_(rt)Y_{r t}=\kappa A_{r t} Z_{t}^{\tilde{\Phi}-1}\left(\frac{K_{t}}{Y_{t}}\right)^{\frac{\alpha(1-\lambda)}{1-\alpha}} L_{r t}Yrt=κArtZtΦ~1(KtYt)α(1λ)1αLrt
where κ κ kappa\kappaκ is a constant, so
log Y r t log Y ¯ r = ( log l r t log l ¯ r ) ( 1 ϕ ~ ) ( log Z t log Z ¯ t ) ( 1 λ ) α 1 α ( log R k t log R ¯ k ) log Y r t log Y ¯ r = log l r t log l ¯ r ( 1 ϕ ~ ) log Z t log Z ¯ t ( 1 λ ) α 1 α log R k t log R ¯ k log Y_(rt)^(')-log bar(Y)_(r)=(log l_(rt)^(')-log bar(l)_(r))-(1- tilde(phi))(log Z_(t)^(')-log bar(Z)_(t))-((1-lambda)alpha)/(1-alpha)(log R_(kt)^(')-log bar(R)_(k))\log Y_{r t}^{\prime}-\log \bar{Y}_{r}=\left(\log l_{r t}^{\prime}-\log \bar{l}_{r}\right)-(1-\tilde{\phi})\left(\log Z_{t}^{\prime}-\log \bar{Z}_{t}\right)-\frac{(1-\lambda) \alpha}{1-\alpha}\left(\log R_{k t}^{\prime}-\log \bar{R}_{k}\right)logYrtlogY¯r=(loglrtlogl¯r)(1ϕ~)(logZtlogZ¯t)(1λ)α1α(logRktlogR¯k).
Long-run changes in aggregate productivity are given as in equation (14) in Proposition 1 where ϕ ~ ϕ ~ tilde(phi)\tilde{\phi}ϕ~ replaces ϕ ϕ phi\phiϕ. A lower value of λ λ lambda\lambdaλ increases ϕ ~ ϕ ~ tilde(phi)\tilde{\phi}ϕ~ and the associated long-term productivity gains from a given increase in research labor. Following the same steps as in the proof of Proposition 2, we obtain the analog of equation (17) for the new path for aggregate productivity, up to a first-order approximation:
log Z t + 1 log Z ¯ t + 1 j = 0 t [ Γ j ( log l r t j log l ¯ r ) ( 1 λ ) α 1 α ( log R k j log R ¯ k ) ] log Z t + 1 log Z ¯ t + 1 j = 0 t Γ j log l r t j log l ¯ r ( 1 λ ) α 1 α log R k j log R ¯ k log Z_(t+1)^(')-log bar(Z)_(t+1)~~sum_(j=0)^(t)[Gamma_(j)(log l_(rt-j)^(')-log bar(l)_(r))-((1-lambda)alpha)/(1-alpha)(log R_(kj)^(')-log bar(R)_(k))]\log Z_{t+1}^{\prime}-\log \bar{Z}_{t+1} \approx \sum_{j=0}^{t}\left[\Gamma_{j}\left(\log l_{r t-j}^{\prime}-\log \bar{l}_{r}\right)-\frac{(1-\lambda) \alpha}{1-\alpha}\left(\log R_{k j}^{\prime}-\log \bar{R}_{k}\right)\right]logZt+1logZ¯t+1j=0t[Γj(loglrtjlogl¯r)(1λ)α1α(logRkjlogR¯k)]
where the decay coefficients are given as in equation (18) with ϕ ~ ϕ ~ tilde(phi)\tilde{\phi}ϕ~ replacing ϕ ϕ phi\phiϕ. The second term on the right-hand side reflects the change in research output Y r t Y r t Y_(rt)Y_{r t}Yrt that results when λ < 1 λ < 1 lambda < 1\lambda<1λ<1 from changes in the capital-output ratio. This second term is equal to zero once the economy converges to the new long-run BGP. The dynamics of aggregate output are given as in Corollary 1.

E. 3 Klette and Kortum (2004) specification

Here we briefly describe our analytic results in a version of our model that follows Klette and Kortum (2004) more closely in assuming a unitary elasticity of substitution between intermediate goods, ρ = 1 ρ = 1 rho=1\rho=1ρ=1. With ρ = 1 ρ = 1 rho=1\rho=1ρ=1, we abstract from growth in the measure of intermediate goods and from innovation by incumbents to improve their own goods.
Specifically, every product that is new to an incumbent firm or an entrant firm is stolen from another incumbent firm (that is, δ m = δ e = 1 δ m = δ e = 1 delta_(m)=delta_(e)=1\delta_{m}=\delta_{e}=1δm=δe=1 ). We normalize the constant total measure of products to 1 , M t = 1 1 , M t = 1 1,M_(t)=11, M_{t}=11,Mt=1. Output of the final good is
(109) log Y t = z log ( y t ( z ) ) M t ( z ) (109) log Y t = z log y t ( z ) M t ( z ) {:(109)log Y_(t)=sum_(z)log(y_(t)(z))M_(t)(z):}\begin{equation*} \log Y_{t}=\sum_{z} \log \left(y_{t}(z)\right) M_{t}(z) \tag{109} \end{equation*}(109)logYt=zlog(yt(z))Mt(z)
For each intermediate good, the markup is fixed at μ μ mu\muμ (determined by the productivity distance between the incumbent producer and a latent competitor). The size of each product (in terms of revenues or input use relative to the total) is s t ( z ) = 1 / M t = 1 s t ( z ) = 1 / M t = 1 s_(t)(z)=1//M_(t)=1s_{t}(z)=1 / M_{t}=1st(z)=1/Mt=1. Substituting the production function (3) into equation (109), and given that the ratio of physical capital to production labor is independent of z z zzz, aggregate output in equilibrium is
Y t = Z t ( K t ) α ( L p t ) 1 α Y t = Z t K t α L p t 1 α Y_(t)=Z_(t)(K_(t))^(alpha)(L_(pt))^(1-alpha)Y_{t}=Z_{t}\left(K_{t}\right)^{\alpha}\left(L_{p t}\right)^{1-\alpha}Yt=Zt(Kt)α(Lpt)1α
where
Z t = exp ( z z M t ( z ) ) Z t = exp z z M t ( z ) Z_(t)=exp(sum_(z)zM_(t)(z))Z_{t}=\exp \left(\sum_{z} z M_{t}(z)\right)Zt=exp(zzMt(z))
With constant markups, variable profits of each product are given by μ 1 μ ( 1 + τ y ) Y t μ 1 μ 1 + τ y Y t (mu-1)/(mu)(1+tau_(y))Y_(t)\frac{\mu-1}{\mu}\left(1+\tau_{y}\right) Y_{t}μ1μ(1+τy)Yt. Given that variable profits are independent of z z zzz, firms have no incentive to use the technology ζ ζ zeta\zetaζ (.) to invest in improvements of the products they own. Without loss of generality, we set x c t = 0 x c t = 0 x_(ct)=0x_{c t}=0xct=0 and ζ ( 0 ) = 1 ζ ( 0 ) = 1 zeta(0)=1\zeta(0)=1ζ(0)=1.
An incumbent firm that owns the right to produce a product with productivity z z zzz possesses the technology to acquire new goods by investing x m t ( z ) x m t ( z ) x_(mt)(z)x_{m t}(z)xmt(z) units of the research good to displace with probability h ( x m t ( z ) ) h x m t ( z ) h(x_(mt)(z))h\left(x_{m t}(z)\right)h(xmt(z)) a product drawn at random from the entire distribution and start producing at t + 1 t + 1 t+1t+1t+1 with a productivity index z z z^(')z^{\prime}z that is a step Δ s > 0 Δ s > 0 Delta_(s) > 0\Delta_{s}>0Δs>0 higher than the stolen product. 32 32 ^(32){ }^{32}32 Similarly, entrants can invest 1 unit of the research good
to displace and improve by Δ S Δ S Delta_(S)\Delta_{S}ΔS a product drawn randomly from the entire distribution. 33 33 ^(33){ }^{33}33
We conjecture (and verify below) that incumbents' investments are independent of z z zzz, so x m t ( z ) = x m t x m t ( z ) = x m t x_(mt)(z)=x_(mt)x_{m t}(z)=x_{m t}xmt(z)=xmt. A firm that owns the rights to produce n n nnn products at time t t ttt invests a total of x m t × n x m t × n x_(mt)xx nx_{m t} \times nxmt×n units of the research to acquire in expectation h ( x m t ) × n h x m t × n h(x_(mt))xx nh\left(x_{m t}\right) \times nh(xmt)×n new products (the expectation of a binomial distribution with parameters h ( x m t ) h x m t h(x_(mt))h\left(x_{m t}\right)h(xmt) and n n nnn ). Note that this innovative technology for incumbents can be equivalently described (as in Klette and Kortum 2004) as an investment of c ( I t / n ) × n c I t / n × n c(I_(t)//n)xx nc\left(I_{t} / n\right) \times nc(It/n)×n to acquire I t × n I t × n I_(t)xx nI_{t} \times nIt×n products in expectation, where c ( ) c ( ) c(*)c(\cdot)c() is increasing and convex. To map these two technologies, we set I t = h ( x m t ) I t = h x m t I_(t)=h(x_(mt))I_{t}=h\left(x_{m t}\right)It=h(xmt) and c ( I t / n ) = x m t c I t / n = x m t c(I_(t)//n)=x_(mt)c\left(I_{t} / n\right)=x_{m t}c(It/n)=xmt.
The measure of incumbent products that are displaced and improved is h ( x m t ) × 1 + h x m t × 1 + h(x_(mt))xx1+h\left(x_{m t}\right) \times 1+h(xmt)×1+ x e t x e t x_(et)x_{e t}xet, and the G G GGG function mapping aggregate investment levels to productivity growth is
G ( x m t , x e t ) = ( h ( x m t ) + x e t ) Δ s G x m t , x e t = h x m t + x e t Δ s G(x_(mt),x_(et))=(h(x_(mt))+x_(et))Delta_(s)G\left(x_{m t}, x_{e t}\right)=\left(h\left(x_{m t}\right)+x_{e t}\right) \Delta_{s}G(xmt,xet)=(h(xmt)+xet)Δs
Impact elasticities are
Θ m G m ( x ¯ m , x ¯ e ) Y ¯ r = h ( x ¯ m ) Δ s Y ¯ r Θ e G e ( x ¯ m , x ¯ e ) Y ¯ r = Δ s Y ¯ r Θ m G m x ¯ m , x ¯ e Y ¯ r = h x ¯ m Δ s Y ¯ r Θ e G e x ¯ m , x ¯ e Y ¯ r = Δ s Y ¯ r {:[Theta_(m)-=G_(m)( bar(x)_(m), bar(x)_(e)) bar(Y)_(r)=h^(')( bar(x)_(m))Delta_(s) bar(Y)_(r)],[Theta_(e)-=G_(e)( bar(x)_(m), bar(x)_(e)) bar(Y)_(r)=Delta_(s) bar(Y)_(r)]:}\begin{gathered} \Theta_{m} \equiv G_{m}\left(\bar{x}_{m}, \bar{x}_{e}\right) \bar{Y}_{r}=h^{\prime}\left(\bar{x}_{m}\right) \Delta_{s} \bar{Y}_{r} \\ \Theta_{e} \equiv G_{e}\left(\bar{x}_{m}, \bar{x}_{e}\right) \bar{Y}_{r}=\Delta_{s} \bar{Y}_{r} \end{gathered}ΘmGm(x¯m,x¯e)Y¯r=h(x¯m)ΔsY¯rΘeGe(x¯m,x¯e)Y¯r=ΔsY¯r
Note that Θ e Θ e Theta_(e)\Theta_{e}Θe can also be written as
(110) Θ e = Δ s x ¯ e Y ¯ r = ( g ¯ Z G ( x ¯ m , 0 ) ) Y ¯ r x ¯ e (110) Θ e = Δ s x ¯ e Y ¯ r = g ¯ Z G x ¯ m , 0 Y ¯ r x ¯ e {:(110)Theta_(e)=Delta_(s) bar(x)_(e) bar(Y)_(r)=( bar(g)_(Z)-G( bar(x)_(m),0))( bar(Y)_(r))/( bar(x)_(e)):}\begin{equation*} \Theta_{e}=\Delta_{s} \bar{x}_{e} \bar{Y}_{r}=\left(\bar{g}_{Z}-G\left(\bar{x}_{m}, 0\right)\right) \frac{\bar{Y}_{r}}{\bar{x}_{e}} \tag{110} \end{equation*}(110)Θe=Δsx¯eY¯r=(g¯ZG(x¯m,0))Y¯rx¯e
which corresponds to the upper bound in equation (37). If the equilibrium allocation of innovative investment is conditionally efficient (i.e., h ( x ¯ m ) = 1 h x ¯ m = 1 h^(')( bar(x)_(m))=1h^{\prime}\left(\bar{x}_{m}\right)=1h(x¯m)=1, so that Θ m = Θ e Θ m = Θ e Theta_(m)=Theta_(e)\Theta_{m}=\Theta_{e}Θm=Θe ), then the impact elasticity Θ Θ Theta\ThetaΘ (for any perturbation to x m t x m t x_(mt)x_{m t}xmt and x e t x e t x_(et)x_{e t}xet ) has an upper bound as given in equation (26) with G ( 0 ) = 0 G ( 0 ) = 0 G(0)=0G(0)=0G(0)=0. Hence, Θ g ¯ Z Θ g ¯ Z Theta <= bar(g)_(Z)\Theta \leq \bar{g}_{Z}Θg¯Z.
The value of a firm that owns the rights to produce n n nnn products (independently of their productivity) is equal to V t n V t n V_(t)nV_{t} nVtn, with
(111) V t = max x m 0 μ 1 μ ( 1 + τ y ) Y t ( 1 τ m t ) P r t x m + 1 1 + R t ( 1 δ c t + h ( x m ) ) V t + 1 (111) V t = max x m 0 μ 1 μ 1 + τ y Y t 1 τ m t P r t x m + 1 1 + R t 1 δ c t + h x m V t + 1 {:(111)V_(t)=max_(x_(m) >= 0)(mu-1)/(mu)(1+tau_(y))Y_(t)-(1-tau_(mt))P_(rt)x_(m)+(1)/(1+R_(t))(1-delta_(ct)+h(x_(m)))V_(t+1):}\begin{equation*} V_{t}=\max _{x_{m} \geq 0} \frac{\mu-1}{\mu}\left(1+\tau_{y}\right) Y_{t}-\left(1-\tau_{m t}\right) P_{r t} x_{m}+\frac{1}{1+R_{t}}\left(1-\delta_{c t}+h\left(x_{m}\right)\right) V_{t+1} \tag{111} \end{equation*}(111)Vt=maxxm0μ1μ(1+τy)Yt(1τmt)Prtxm+11+Rt(1δct+h(xm))Vt+1
where the displacement probability of each product, δ c t δ c t delta_(ct)\delta_{c t}δct, is taken as given by the firm and, in equilibrium, is equal to δ c t = h ( x m t ) + x e t δ c t = h x m t + x e t delta_(ct)=h(x_(mt))+x_(et)\delta_{c t}=h\left(x_{m t}\right)+x_{e t}δct=h(xmt)+xet. The first-order condition of this profit
maximization implies that x m t ( z ) = x m t x m t ( z ) = x m t x_(mt)(z)=x_(mt)x_{m t}(z)=x_{m t}xmt(z)=xmt with
(112) ( 1 τ m t ) P r t 1 1 + R t h ( x m t ) V t + 1 (112) 1 τ m t P r t 1 1 + R t h x m t V t + 1 {:(112)(1-tau_(mt))P_(rt) >= (1)/(1+R_(t))h^(')(x_(mt))V_(t+1):}\begin{equation*} \left(1-\tau_{m t}\right) P_{r t} \geq \frac{1}{1+R_{t}} h^{\prime}\left(x_{m t}\right) V_{t+1} \tag{112} \end{equation*}(112)(1τmt)Prt11+Rth(xmt)Vt+1
confirming that x m t x m t x_(mt)x_{m t}xmt is independent of z z zzz. The free entry condition is
(113) ( 1 τ e t ) P r t 1 1 + R t V t + 1 (113) 1 τ e t P r t 1 1 + R t V t + 1 {:(113)(1-tau_(et))P_(rt) >= (1)/(1+R_(t))V_(t+1):}\begin{equation*} \left(1-\tau_{e t}\right) P_{r t} \geq \frac{1}{1+R_{t}} V_{t+1} \tag{113} \end{equation*}(113)(1τet)Prt11+RtVt+1
The equilibrium allocation of innovative investment is conditionally efficient ( h ( x ¯ m ) = h x ¯ m = (h^(')( bar(x)_(m))=:}\left(h^{\prime}\left(\bar{x}_{m}\right)=\right.(h(x¯m)= 1) if and only if innovation policies are uniform ( τ m t = τ e t ) τ m t = τ e t (tau_(mt)=tau_(et))\left(\tau_{m t}=\tau_{e t}\right)(τmt=τet).
Finally, we show how we can measure P ¯ r x ¯ e ( 1 + τ y ) Y ¯ t P ¯ r x ¯ e 1 + τ y Y ¯ t ( bar(P)_(r) bar(x)_(e))/((1+tau_(y)) bar(Y)_(t))\frac{\bar{P}_{r} \bar{x}_{e}}{\left(1+\tau_{y}\right) \bar{Y}_{t}}P¯rx¯e(1+τy)Y¯t, which is required to calculate Θ e Θ e Theta_(e)\Theta_{e}Θe in equation (110). Defining v t = V t / ( ( 1 + τ y ) Y t ) v t = V t / 1 + τ y Y t v_(t)=V_(t)//((1+tau_(y))Y_(t))v_{t}=V_{t} /\left(\left(1+\tau_{y}\right) Y_{t}\right)vt=Vt/((1+τy)Yt) and using the fact that, with ρ = 1 ρ = 1 rho=1\rho=1ρ=1 and g M t = 0 , s e t + 1 = f e t + 1 = x e t g M t = 0 , s e t + 1 = f e t + 1 = x e t g_(Mt)=0,s_(et+1)=f_(et+1)=x_(et)g_{M t}=0, s_{e t+1}=f_{e t+1}=x_{e t}gMt=0,set+1=fet+1=xet, we have from equation (113),

where by equation (111),
v ¯ = [ μ 1 μ ( 1 τ ¯ m ) P ¯ r x ¯ m ( 1 + τ y ) Y ¯ t ] 1 exp ( g ¯ Y ) 1 + R ¯ ( 1 s ¯ e ) v ¯ = μ 1 μ 1 τ ¯ m P ¯ r x ¯ m 1 + τ y Y ¯ t 1 exp g ¯ Y 1 + R ¯ 1 s ¯ e bar(v)=([(mu-1)/(mu)-(1- bar(tau)_(m))( bar(P)_(r) bar(x)_(m))/((1+tau_(y)) bar(Y)_(t))])/(1-(exp( bar(g)_(Y)))/(1+( bar(R)))(1- bar(s)_(e)))\bar{v}=\frac{\left[\frac{\mu-1}{\mu}-\left(1-\bar{\tau}_{m}\right) \frac{\bar{P}_{r} \bar{x}_{m}}{\left(1+\tau_{y}\right) \bar{Y}_{t}}\right]}{1-\frac{\exp \left(\bar{g}_{Y}\right)}{1+\bar{R}}\left(1-\bar{s}_{e}\right)}v¯=[μ1μ(1τ¯m)P¯rx¯m(1+τy)Y¯t]1exp(g¯Y)1+R¯(1s¯e)
which is analogous to equation (90) in the baseline model.

E. 4 Alternative specification of incumbent technology to acquire new products

In this section, we consider a specification of the investment technology for incumbent firms to acquire new products (which is nested in the generalized model of Akcigit and Kerr 2018) in which investments to acquire new products do not scale up one to one with firm size, but instead scale up with the number of products that the firm has. These two specifications are equivalent when ρ = 1 ρ = 1 rho=1\rho=1ρ=1, as discussed in Section E.3.
We assume that a firm with a product with productivity z z zzz has a technology to acquire a new product with probability h ( x m t ( z ) M t ) h x m t ( z ) M t h(x_(mt)(z)M_(t))h\left(x_{m t}(z) M_{t}\right)h(xmt(z)Mt) (as opposed to h ( x m t ( z ) / s t ( z ) ) h x m t ( z ) / s t ( z ) h(x_(mt)(z)//s_(t)(z))h\left(x_{m t}(z) / s_{t}(z)\right)h(xmt(z)/st(z)) in our baseline model). With probability δ m δ m delta_(m)\delta_{m}δm it displaces a product drawn at random from the entire distribution (as opposed to displacing a product with productivity z z zzz ), so that the displaced product has productivity index z z zzz at time t t ttt with E z ρ 1 = Z t ρ 1 / M t E z ρ 1 = Z t ρ 1 / M t Ez^(rho-1)=Z_(t)^(rho-1)//M_(t)\mathbb{E} z^{\rho-1}=Z_{t}^{\rho-1} / M_{t}Ezρ1=Ztρ1/Mt. The
incumbent firm that stole this product can produce it at t + 1 t + 1 t+1t+1t+1 with a new productivity index z z z^(')z^{\prime}z such that the expected value of the term z ρ 1 z ρ 1 z^('rho-1)z^{\prime \rho-1}zρ1 is equal to E z ρ 1 = η m s Z t ρ 1 / M t E z ρ 1 = η m s Z t ρ 1 / M t Ez^('rho-1)=eta_(ms)Z_(t)^(rho-1)//M_(t)\mathbb{E} z^{\prime \rho-1}=\eta_{m s} Z_{t}^{\rho-1} / M_{t}Ezρ1=ηmsZtρ1/Mt (instead of η m s z ρ 1 η m s z ρ 1 eta_(ms)z^(rho-1)\eta_{m s} z^{\rho-1}ηmszρ1 in the baseline model), with η m s > 1 η m s > 1 eta_(ms) > 1\eta_{m s}>1ηms>1. With complementary probability 1 δ m 1 δ m 1-delta_(m)1-\delta_{m}1δm the newly acquired product is new to society, with productivity index z z z^(')z^{\prime}z drawn from a distribution such that the expected value of the term z ρ ρ 1 z ρ ρ 1 z^(rho rho-1)z^{\rho \rho-1}zρρ1 is equal to E z ρ 1 = η m n Z t ρ 1 / M t E z ρ 1 = η m n Z t ρ 1 / M t Ez^('rho-1)=eta_(mn)Z_(t)^(rho-1)//M_(t)\mathbb{E} z^{\prime \rho-1}=\eta_{m n} Z_{t}^{\rho-1} / M_{t}Ezρ1=ηmnZtρ1/Mt (instead of η m n z ρ 1 η m n z ρ 1 eta_(mn)z^(rho-1)\eta_{m n} z^{\rho-1}ηmnzρ1 in the baseline model), with η m n > 0 η m n > 0 eta_(mn) > 0\eta_{m n}>0ηmn>0. We define η m = δ m η m s + ( 1 δ m ) η m n η m = δ m η m s + 1 δ m η m n eta_(m)=delta_(m)eta_(ms)+(1-delta_(m))eta_(mn)\eta_{m}=\delta_{m} \eta_{m s}+\left(1-\delta_{m}\right) \eta_{m n}ηm=δmηms+(1δm)ηmn. The investment technologies of entrants and of incumbent firms to improve continuing products remain unchanged.
We conjecture (and verify below) that the level of investment by incumbent firms in acquiring new products is independent of the productivity index z z zzz of the product with which this investment is associated. So, x m t ( z ) = x m t / M t x m t ( z ) = x m t / M t x_(mt)(z)=x_(mt)//M_(t)x_{m t}(z)=x_{m t} / M_{t}xmt(z)=xmt/Mt (instead of x m t ( z ) = s t ( z ) x m t x m t ( z ) = s t ( z ) x m t x_(mt)(z)=s_(t)(z)x_(mt)x_{m t}(z)=s_{t}(z) x_{m t}xmt(z)=st(z)xmt in the baseline model). We also conjecture that investments by incumbent firms in improving their continuing products scale up with the size of the product with which this investment is associated, as in the baseline model. With x m t ( z ) = x m t / M t x m t ( z ) = x m t / M t x_(mt)(z)=x_(mt)//M_(t)x_{m t}(z)=x_{m t} / M_{t}xmt(z)=xmt/Mt and x c t ( z ) = x c t ( z ) = x_(ct)(z)=x_{c t}(z)=xct(z)= x c t s t ( z ) x c t s t ( z ) x_(ct)s_(t)(z)x_{c t} s_{t}(z)xctst(z), a firm that owns the right to produce n n nnn products with productivities { z 1 , , z n } z 1 , , z n {z_(1),dots,z_(n)}\left\{z_{1}, \ldots, z_{n}\right\}{z1,,zn} has size s t = i = 1 n s t ( z i ) s t = i = 1 n s t z i s_(t)=sum_(i=1)^(n)s_(t)(z_(i))s_{t}=\sum_{i=1}^{n} s_{t}\left(z_{i}\right)st=i=1nst(zi) and spends a total r t = s t ( x c t + x m t M s t ) r t = s t x c t + x m t M s t r_(t)=s_(t)(x_(ct)+(x_(mt))/(M)s_(t))r_{t}=s_{t}\left(x_{c t}+\frac{x_{m t}}{M} s_{t}\right)rt=st(xct+xmtMst) on the research good. Assuming a positive cross-firm correlation between firm size s t s t s_(t)s_{t}st and average product size s t / n s t / n s_(t)//ns_{t} / nst/n, this alternative specification generates a negative correlation between firm size s t s t s_(t)s_{t}st and innovation intensity r t s t = x c t + x m t M n s t r t s t = x c t + x m t M n s t (r_(t))/(s_(t))=x_(ct)+(x_(mt))/(M)(n)/(s_(t))\frac{r_{t}}{s_{t}}=x_{c t}+\frac{x_{m t}}{M} \frac{n}{s_{t}}rtst=xct+xmtMnst. Similarly, since small firms have products with a lower average z z zzz, their investments in acquiring new products produce mean reversion in the average value of z z zzz (since the new z z z^(')z^{\prime}z s they acquire are drawn from the entire distribution and not from products with productivity z z zzz as in the baseline model), implying higher growth relative to firms with a higher average value of z . 34 z . 34 z.^(34)z .{ }^{34}z.34 In what follows, we discuss how our analytic results are affected under the modified technology.
Incumbent firms acquire in the aggregate a measure h ( x m t ) M t h x m t M t h(x_(mt))M_(t)h\left(x_{m t}\right) M_{t}h(xmt)Mt new products - a measure δ m h ( x m t ) M t δ m h x m t M t delta_(m)h(x_(mt))M_(t)\delta_{m} h\left(x_{m t}\right) M_{t}δmh(xmt)Mt are stolen from other incumbent firms and complementary measure ( 1 δ m ) h ( x m t ) M t 1 δ m h x m t M t (1-delta_(m))h(x_(mt))M_(t)\left(1-\delta_{m}\right) h\left(x_{m t}\right) M_{t}(1δm)h(xmt)Mt are new to society. The average value of z ρ 1 z ρ 1 z^('rho-1)z^{\prime \rho-1}zρ1 across all newly acquired products in incumbent firms at t + 1 t + 1 t+1t+1t+1 is η m Z t ρ 1 M t η m Z t ρ 1 M t eta_(m)(Z_(t)^(rho-1))/(M_(t))\eta_{m} \frac{\mathrm{Z}_{t}^{\rho-1}}{M_{t}}ηmZtρ1Mt. With x c t ( z ) = s t ( z ) x c t x c t ( z ) = s t ( z ) x c t x_(ct)(z)=s_(t)(z)x_(ct)x_{c t}(z)=s_{t}(z) x_{c t}xct(z)=st(z)xct, and following the logic of the proof of Lemma 3, we obtain the same law of motion for aggregate productivity (28) and resource constraint of the research good (27) as in our baseline model. Therefore, Θ c , Θ m Θ c , Θ m Theta_(c),Theta_(m)\Theta_{c}, \Theta_{m}Θc,Θm and Θ e Θ e Theta_(e)\Theta_{e}Θe are given as in equations (66)-(68) as well as equation (36) in
our baseline model.
We now verify our conjecture that in equilibrium, firms choose x m t ( z ) = x m t / M t x m t ( z ) = x m t / M t x_(mt)(z)=x_(mt)//M_(t)x_{m t}(z)=x_{m t} / M_{t}xmt(z)=xmt/Mt and x c t ( z ) = s t ( z ) x c t x c t ( z ) = s t ( z ) x c t x_(ct)(z)=s_(t)(z)x_(ct)x_{c t}(z)=s_{t}(z) x_{c t}xct(z)=st(z)xct. Under this conjecture, the value of a continuing product with productivity index z z zzz for an incumbent firm is given by V t ( z ) + U t V t ( z ) + U t V_(t)(z)+U_(t)V_{t}(z)+U_{t}Vt(z)+Ut where V t ( z ) V t ( z ) V_(t)(z)V_{t}(z)Vt(z) denotes the discounted present value for an incumbent firm of the dividends associated with a product with productivity z z zzz at time t t ttt, and U t U t U_(t)U_{t}Ut denotes the value for an incumbent firm of this product in facilitating further acquisition of products (which, in contrast to our baseline model, is independent of z z zzz ). The value of a firm that owns the technology to produce n n nnn products with productivities { z 1 , , z n } z 1 , , z n {z_(1),dots,z_(n)}\left\{z_{1}, \ldots, z_{n}\right\}{z1,,zn} is equal to
V t ( z ) i = 1 n s t ( z i ) + U t n V t ( z ) i = 1 n s t z i + U t n V_(t)(z)sum_(i=1)^(n)s_(t)(z_(i))+U_(t)nV_{t}(z) \sum_{i=1}^{n} s_{t}\left(z_{i}\right)+U_{t} nVt(z)i=1nst(zi)+Utn
Specifically, V t ( z ) = s t ( z ) V t V t ( z ) = s t ( z ) V t V_(t)(z)=s_(t)(z)V_(t)V_{t}(z)=s_{t}(z) V_{t}Vt(z)=st(z)Vt with
(114) V t = max x c 0 ( 1 + τ y ) μ 1 μ Y t ( 1 τ c t ) P r t x c + 1 1 + R t V t + 1 ( 1 δ c t ) ζ ( x c ) Z t ρ 1 Z t + 1 ρ 1 (114) V t = max x c 0 1 + τ y μ 1 μ Y t 1 τ c t P r t x c + 1 1 + R t V t + 1 1 δ c t ζ x c Z t ρ 1 Z t + 1 ρ 1 {:(114)V_(t)=max_(x_(c) >= 0)(1+tau_(y))(mu-1)/(mu)Y_(t)-(1-tau_(ct))P_(rt)x_(c)+(1)/(1+R_(t))V_(t+1)(1-delta_(ct))zeta(x_(c))(Z_(t)^(rho-1))/(Z_(t+1)^(rho-1)):}\begin{equation*} V_{t}=\max _{x_{c} \geq 0}\left(1+\tau_{y}\right) \frac{\mu-1}{\mu} Y_{t}-\left(1-\tau_{c t}\right) P_{r t} x_{c}+\frac{1}{1+R_{t}} V_{t+1}\left(1-\delta_{c t}\right) \zeta\left(x_{c}\right) \frac{Z_{t}^{\rho-1}}{Z_{t+1}^{\rho-1}} \tag{114} \end{equation*}(114)Vt=maxxc0(1+τy)μ1μYt(1τct)Prtxc+11+RtVt+1(1δct)ζ(xc)Ztρ1Zt+1ρ1
and
U t = max x m 0 ( 1 τ m t ) P r t x m + 1 1 + R t ( h ( x m M t ) V t + 1 η m Z t ρ 1 M t Z t + 1 ρ 1 + ( 1 δ c t + h ( x m M t ) ) U t + 1 ) U t = max x m 0 1 τ m t P r t x m + 1 1 + R t h x m M t V t + 1 η m Z t ρ 1 M t Z t + 1 ρ 1 + 1 δ c t + h x m M t U t + 1 U_(t)=max_(x_(m) >= 0)-(1-tau_(mt))P_(rt)x_(m)+(1)/(1+R_(t))(h(x_(m)M_(t))V_(t+1)eta_(m)(Z_(t)^(rho-1))/(M_(t)Z_(t+1)^(rho-1))+(1-delta_(ct)+h(x_(m)M_(t)))U_(t+1))U_{t}=\max _{x_{m} \geq 0}-\left(1-\tau_{m t}\right) P_{r t} x_{m}+\frac{1}{1+R_{t}}\left(h\left(x_{m} M_{t}\right) V_{t+1} \eta_{m} \frac{Z_{t}^{\rho-1}}{M_{t} Z_{t+1}^{\rho-1}}+\left(1-\delta_{c t}+h\left(x_{m} M_{t}\right)\right) U_{t+1}\right)Ut=maxxm0(1τmt)Prtxm+11+Rt(h(xmMt)Vt+1ηmZtρ1MtZt+1ρ1+(1δct+h(xmMt))Ut+1)
The first-order condition of the incumbent firm's profit maximization problem (114) with respect to x c x c x_(c)x_{c}xc implies that x c t ( z ) = s t ( z ) x c t x c t ( z ) = s t ( z ) x c t x_(ct)(z)=s_(t)(z)x_(ct)x_{c t}(z)=s_{t}(z) x_{c t}xct(z)=st(z)xct with
(116) ( 1 τ c t ) P r t 1 1 + R t V t + 1 ( 1 δ c t ) ζ ( x c t ) Z t ρ 1 Z t + 1 ρ 1 (116) 1 τ c t P r t 1 1 + R t V t + 1 1 δ c t ζ x c t Z t ρ 1 Z t + 1 ρ 1 {:(116)(1-tau_(ct))P_(rt) >= (1)/(1+R_(t))V_(t+1)(1-delta_(ct))zeta^(')(x_(ct))(Z_(t)^(rho-1))/(Z_(t+1)^(rho-1)):}\begin{equation*} \left(1-\tau_{c t}\right) P_{r t} \geq \frac{1}{1+R_{t}} V_{t+1}\left(1-\delta_{c t}\right) \zeta^{\prime}\left(x_{c t}\right) \frac{Z_{t}^{\rho-1}}{Z_{t+1}^{\rho-1}} \tag{116} \end{equation*}(116)(1τct)Prt11+RtVt+1(1δct)ζ(xct)Ztρ1Zt+1ρ1
and the first-order condition of the incumbent firm's profit maximization problem (115) with respect to x m x m x_(m)x_{m}xm implies that x m t ( z ) = x m t / M t x m t ( z ) = x m t / M t x_(mt)(z)=x_(mt)//M_(t)x_{m t}(z)=x_{m t} / M_{t}xmt(z)=xmt/Mt with
(117) ( 1 τ m t ) P r t 1 1 + R t h ( x m t ) ( V t + 1 η m Z t ρ 1 Z t + 1 ρ 1 + U t + 1 M t ) (117) 1 τ m t P r t 1 1 + R t h x m t V t + 1 η m Z t ρ 1 Z t + 1 ρ 1 + U t + 1 M t {:(117)(1-tau_(mt))P_(rt) >= (1)/(1+R_(t))h^(')(x_(mt))(V_(t+1)eta_(m)(Z_(t)^(rho-1))/(Z_(t+1)^(rho-1))+U_(t+1)M_(t)):}\begin{equation*} \left(1-\tau_{m t}\right) P_{r t} \geq \frac{1}{1+R_{t}} h^{\prime}\left(x_{m t}\right)\left(V_{t+1} \eta_{m} \frac{Z_{t}^{\rho-1}}{Z_{t+1}^{\rho-1}}+U_{t+1} M_{t}\right) \tag{117} \end{equation*}(117)(1τmt)Prt11+Rth(xmt)(Vt+1ηmZtρ1Zt+1ρ1+Ut+1Mt)
The free entry condition is
(118) ( 1 τ e t ) P r t 1 1 + R t ( V t + 1 η e Z t ρ 1 Z t + 1 ρ 1 + U t + 1 M t ) (118) 1 τ e t P r t 1 1 + R t V t + 1 η e Z t ρ 1 Z t + 1 ρ 1 + U t + 1 M t {:(118)(1-tau_(et))P_(rt) >= (1)/(1+R_(t))(V_(t+1)eta_(e)(Z_(t)^(rho-1))/(Z_(t+1)^(rho-1))+U_(t+1)M_(t)):}\begin{equation*} \left(1-\tau_{e t}\right) P_{r t} \geq \frac{1}{1+R_{t}}\left(V_{t+1} \eta_{e} \frac{Z_{t}^{\rho-1}}{Z_{t+1}^{\rho-1}}+U_{t+1} M_{t}\right) \tag{118} \end{equation*}(118)(1τet)Prt11+Rt(Vt+1ηeZtρ1Zt+1ρ1+Ut+1Mt)
These first-order conditions are equalities if x c t , x m t > 0 x c t , x m t > 0 x_(ct),x_(mt) > 0x_{c t}, x_{m t}>0xct,xmt>0.
Defining v t = V t / ( ( 1 + τ y ) Y t ) v t = V t / 1 + τ y Y t v_(t)=V_(t)//((1+tau_(y))Y_(t))v_{t}=V_{t} /\left(\left(1+\tau_{y}\right) Y_{t}\right)vt=Vt/((1+τy)Yt) and u t = U t M t / ( ( 1 + τ y ) Y t ) u t = U t M t / 1 + τ y Y t u_(t)=U_(t)M_(t)//((1+tau_(y))Y_(t))u_{t}=U_{t} M_{t} /\left(\left(1+\tau_{y}\right) Y_{t}\right)ut=UtMt/((1+τy)Yt), and assuming an interior equilibrium, we can rewrite this system of equations as
(119) ( 1 τ c t ) P r t ( 1 τ y ) Y t = Y t + 1 / Y t 1 + R t v t + 1 ( 1 δ c t ) ζ ( x c t ) Z t ρ 1 Z t + 1 ρ 1 (120) ( 1 τ m t ) P r t ( 1 τ y ) Y t = Y t + 1 / Y t 1 + R t h ( x m t ) ( v t + 1 η m Z t ρ 1 Z t + 1 ρ 1 + u t + 1 M t M t + 1 ) (121) ( 1 τ e t ) P r t ( 1 τ y ) Y t = Y t + 1 / Y t 1 + R t ( v t + 1 η e Z t ρ 1 Z t + 1 ρ 1 + u t + 1 M t M t + 1 ) v t = μ 1 μ ( 1 τ c t ) P r t x c t ( 1 τ y ) Y t + Y t + 1 / Y t 1 + R t v t + 1 ( 1 δ c t ) ζ ( x c t ) Z t ρ 1 Z t + 1 ρ 1 u t = ( 1 τ m t ) P r t x m t ( 1 + τ y ) Y t + Y t + 1 / Y t 1 + R t ( h ( x m t ) v t + 1 η m Z t ρ 1 Z t + 1 ρ 1 + ( 1 δ c t + h ( x m t ) ) u t + 1 M t M t + 1 ) (119) 1 τ c t P r t 1 τ y Y t = Y t + 1 / Y t 1 + R t v t + 1 1 δ c t ζ x c t Z t ρ 1 Z t + 1 ρ 1 (120) 1 τ m t P r t 1 τ y Y t = Y t + 1 / Y t 1 + R t h x m t v t + 1 η m Z t ρ 1 Z t + 1 ρ 1 + u t + 1 M t M t + 1 (121) 1 τ e t P r t 1 τ y Y t = Y t + 1 / Y t 1 + R t v t + 1 η e Z t ρ 1 Z t + 1 ρ 1 + u t + 1 M t M t + 1 v t = μ 1 μ 1 τ c t P r t x c t 1 τ y Y t + Y t + 1 / Y t 1 + R t v t + 1 1 δ c t ζ x c t Z t ρ 1 Z t + 1 ρ 1 u t = 1 τ m t P r t x m t 1 + τ y Y t + Y t + 1 / Y t 1 + R t h x m t v t + 1 η m Z t ρ 1 Z t + 1 ρ 1 + 1 δ c t + h x m t u t + 1 M t M t + 1 {:[(119)((1-tau_(ct))P_(rt))/((1-tau_(y))Y_(t))=(Y_(t+1)//Y_(t))/(1+R_(t))v_(t+1)(1-delta_(ct))zeta^(')(x_(ct))(Z_(t)^(rho-1))/(Z_(t+1)^(rho-1))],[(120)((1-tau_(mt))P_(rt))/((1-tau_(y))Y_(t))=(Y_(t+1)//Y_(t))/(1+R_(t))h^(')(x_(mt))(v_(t+1)eta_(m)(Z_(t)^(rho-1))/(Z_(t+1)^(rho-1))+u_(t+1)(M_(t))/(M_(t+1)))],[(121)((1-tau_(et))P_(rt))/((1-tau_(y))Y_(t))=(Y_(t+1)//Y_(t))/(1+R_(t))(v_(t+1)eta_(e)(Z_(t)^(rho-1))/(Z_(t+1)^(rho-1))+u_(t+1)(M_(t))/(M_(t+1)))],[v_(t)=(mu-1)/(mu)-((1-tau_(ct))P_(rt)x_(ct))/((1-tau_(y))Y_(t))+(Y_(t+1)//Y_(t))/(1+R_(t))v_(t+1)(1-delta_(ct))zeta(x_(ct))(Z_(t)^(rho-1))/(Z_(t+1)^(rho-1))],[u_(t)=-((1-tau_(mt))P_(rt)x_(mt))/((1+tau_(y))Y_(t))+(Y_(t+1)//Y_(t))/(1+R_(t))(h(x_(mt))v_(t+1)eta_(m)(Z_(t)^(rho-1))/(Z_(t+1)^(rho-1))+(1-delta_(ct)+h(x_(mt)))u_(t+1)(M_(t))/(M_(t+1)))]:}\begin{gather*} \frac{\left(1-\tau_{c t}\right) P_{r t}}{\left(1-\tau_{y}\right) Y_{t}}=\frac{Y_{t+1} / Y_{t}}{1+R_{t}} v_{t+1}\left(1-\delta_{c t}\right) \zeta^{\prime}\left(x_{c t}\right) \frac{Z_{t}^{\rho-1}}{Z_{t+1}^{\rho-1}} \tag{119}\\ \frac{\left(1-\tau_{m t}\right) P_{r t}}{\left(1-\tau_{y}\right) Y_{t}}=\frac{Y_{t+1} / Y_{t}}{1+R_{t}} h^{\prime}\left(x_{m t}\right)\left(v_{t+1} \eta_{m} \frac{Z_{t}^{\rho-1}}{Z_{t+1}^{\rho-1}}+u_{t+1} \frac{M_{t}}{M_{t+1}}\right) \tag{120}\\ \frac{\left(1-\tau_{e t}\right) P_{r t}}{\left(1-\tau_{y}\right) Y_{t}}=\frac{Y_{t+1} / Y_{t}}{1+R_{t}}\left(v_{t+1} \eta_{e} \frac{Z_{t}^{\rho-1}}{Z_{t+1}^{\rho-1}}+u_{t+1} \frac{M_{t}}{M_{t+1}}\right) \tag{121}\\ v_{t}=\frac{\mu-1}{\mu}-\frac{\left(1-\tau_{c t}\right) P_{r t} x_{c t}}{\left(1-\tau_{y}\right) Y_{t}}+\frac{Y_{t+1} / Y_{t}}{1+R_{t}} v_{t+1}\left(1-\delta_{c t}\right) \zeta\left(x_{c t}\right) \frac{Z_{t}^{\rho-1}}{Z_{t+1}^{\rho-1}} \\ u_{t}=-\frac{\left(1-\tau_{m t}\right) P_{r t} x_{m t}}{\left(1+\tau_{y}\right) Y_{t}}+\frac{Y_{t+1} / Y_{t}}{1+R_{t}}\left(h\left(x_{m t}\right) v_{t+1} \eta_{m} \frac{Z_{t}^{\rho-1}}{Z_{t+1}^{\rho-1}}+\left(1-\delta_{c t}+h\left(x_{m t}\right)\right) u_{t+1} \frac{M_{t}}{M_{t+1}}\right) \end{gather*}(119)(1τct)Prt(1τy)Yt=Yt+1/Yt1+Rtvt+1(1δct)ζ(xct)Ztρ1Zt+1ρ1(120)(1τmt)Prt(1τy)Yt=Yt+1/Yt1+Rth(xmt)(vt+1ηmZtρ1Zt+1ρ1+ut+1MtMt+1)(121)(1τet)Prt(1τy)Yt=Yt+1/Yt1+Rt(vt+1ηeZtρ1Zt+1ρ1+ut+1MtMt+1)vt=μ1μ(1τct)Prtxct(1τy)Yt+Yt+1/Yt1+Rtvt+1(1δct)ζ(xct)Ztρ1Zt+1ρ1ut=(1τmt)Prtxmt(1+τy)Yt+Yt+1/Yt1+Rt(h(xmt)vt+1ηmZtρ1Zt+1ρ1+(1δct+h(xmt))ut+1MtMt+1)
Recall that in the baseline model, if new innovation policies satisfy condition (32), for a given value of Y r t Y r t Y_(rt)Y_{r t}Yrt we are able to solve for x c t , x m t x c t , x m t x_(ct),x_(mt)x_{c t}, x_{m t}xct,xmt, and x e t x e t x_(et)x_{e t}xet as a static system of equations using equations (64), (65), and the resource constraint (27). In contrast, in this alternative model specification, we must also solve for v t + 1 v t + 1 v_(t+1)v_{t+1}vt+1 and u t + 1 u t + 1 u_(t+1)u_{t+1}ut+1, unless x c t x c t x_(ct)x_{c t}xct is fixed exogenously ( x c t = x ¯ c ) x c t = x ¯ c (x_(ct)= bar(x)_(c))\left(x_{c t}=\bar{x}_{c}\right)(xct=x¯c) and η m = η e η m = η e eta_(m)=eta_(e)\eta_{m}=\eta_{e}ηm=ηe. In this case, x m t x m t x_(mt)x_{m t}xmt and x e t x e t x_(et)x_{e t}xet can be solved using the two following static equations:
1 τ m t 1 τ e t = h ( x m t ) and x ¯ c + x m t + x e t = Y r t 1 τ m t 1 τ e t = h x m t  and  x ¯ c + x m t + x e t = Y r t (1-tau_(mt))/(1-tau_(et))=h^(')(x_(mt))" and " bar(x)_(c)+x_(mt)+x_(et)=Y_(rt)\frac{1-\tau_{m t}}{1-\tau_{e t}}=h^{\prime}\left(x_{m t}\right) \text { and } \bar{x}_{c}+x_{m t}+x_{e t}=Y_{r t}1τmt1τet=h(xmt) and x¯c+xmt+xet=Yrt
without requiring to solve for v t + 1 v t + 1 v_(t+1)v_{t+1}vt+1 and u t + 1 u t + 1 u_(t+1)u_{t+1}ut+1.
In the BGP, x c t = x ¯ c , x m t = x ¯ m , x e t = x ¯ e , v t = v ¯ x c t = x ¯ c , x m t = x ¯ m , x e t = x ¯ e , v t = v ¯ x_(ct)= bar(x)_(c),x_(mt)= bar(x)_(m),x_(et)= bar(x)_(e),v_(t)= bar(v)x_{c t}=\bar{x}_{c}, x_{m t}=\bar{x}_{m}, x_{e t}=\bar{x}_{e}, v_{t}=\bar{v}xct=x¯c,xmt=x¯m,xet=x¯e,vt=v¯, and u t = u ¯ u t = u ¯ u_(t)= bar(u)u_{t}=\bar{u}ut=u¯. The first-order conditions (119), (121) in the BGP are
(122) ( 1 τ c ) P ¯ r t ( 1 + τ y ) Y ¯ t = exp ( g ¯ Y ( ρ 1 ) g ¯ Z ) 1 + R ¯ v ¯ ( 1 δ ¯ c ) ζ ( x ¯ c ) (123) ( 1 τ m ) P ¯ r t ( 1 + τ y ) Y ¯ t = exp ( g ¯ Y ( ρ 1 ) g ¯ Z ) 1 + R ¯ h ( x ¯ m ) ( v ¯ η m + u ¯ exp ( ( ρ 1 ) g ¯ Z g ¯ M ) ) (124) ( 1 τ e ) P ¯ r t ( 1 + τ y ) Y ¯ t = exp ( g ¯ Y ( ρ 1 ) g ¯ Z ) 1 + R ¯ ( v ¯ η e + u ¯ exp ( ( ρ 1 ) g ¯ Z g ¯ M ) ) (122) 1 τ c P ¯ r t 1 + τ y Y ¯ t = exp g ¯ Y ( ρ 1 ) g ¯ Z 1 + R ¯ v ¯ 1 δ ¯ c ζ x ¯ c (123) 1 τ m P ¯ r t 1 + τ y Y ¯ t = exp g ¯ Y ( ρ 1 ) g ¯ Z 1 + R ¯ h x ¯ m v ¯ η m + u ¯ exp ( ρ 1 ) g ¯ Z g ¯ M (124) 1 τ e P ¯ r t 1 + τ y Y ¯ t = exp g ¯ Y ( ρ 1 ) g ¯ Z 1 + R ¯ v ¯ η e + u ¯ exp ( ρ 1 ) g ¯ Z g ¯ M {:[(122)((1-tau_(c)) bar(P)_(rt))/((1+tau_(y)) bar(Y)_(t))=(exp( bar(g)_(Y)-(rho-1) bar(g)_(Z)))/(1+( bar(R))) bar(v)(1- bar(delta)_(c))zeta^(')( bar(x)_(c))],[(123)((1-tau_(m)) bar(P)_(rt))/((1+tau_(y)) bar(Y)_(t))=(exp( bar(g)_(Y)-(rho-1) bar(g)_(Z)))/(1+( bar(R)))h^(')( bar(x)_(m))(( bar(v))eta_(m)+( bar(u))exp((rho-1) bar(g)_(Z)- bar(g)_(M)))],[(124)((1-tau_(e)) bar(P)_(rt))/((1+tau_(y)) bar(Y)_(t))=(exp( bar(g)_(Y)-(rho-1) bar(g)_(Z)))/(1+( bar(R)))(( bar(v))eta_(e)+( bar(u))exp((rho-1) bar(g)_(Z)- bar(g)_(M)))]:}\begin{gather*} \frac{\left(1-\tau_{c}\right) \bar{P}_{r t}}{\left(1+\tau_{y}\right) \bar{Y}_{t}}=\frac{\exp \left(\bar{g}_{Y}-(\rho-1) \bar{g}_{Z}\right)}{1+\bar{R}} \bar{v}\left(1-\bar{\delta}_{c}\right) \zeta^{\prime}\left(\bar{x}_{c}\right) \tag{122}\\ \frac{\left(1-\tau_{m}\right) \bar{P}_{r t}}{\left(1+\tau_{y}\right) \bar{Y}_{t}}=\frac{\exp \left(\bar{g}_{Y}-(\rho-1) \bar{g}_{Z}\right)}{1+\bar{R}} h^{\prime}\left(\bar{x}_{m}\right)\left(\bar{v} \eta_{m}+\bar{u} \exp \left((\rho-1) \bar{g}_{Z}-\bar{g}_{M}\right)\right) \tag{123}\\ \frac{\left(1-\tau_{e}\right) \bar{P}_{r t}}{\left(1+\tau_{y}\right) \bar{Y}_{t}}=\frac{\exp \left(\bar{g}_{Y}-(\rho-1) \bar{g}_{Z}\right)}{1+\bar{R}}\left(\bar{v} \eta_{e}+\bar{u} \exp \left((\rho-1) \bar{g}_{Z}-\bar{g}_{M}\right)\right) \tag{124} \end{gather*}(122)(1τc)P¯rt(1+τy)Y¯t=exp(g¯Y(ρ1)g¯Z)1+R¯v¯(1δ¯c)ζ(x¯c)(123)(1τm)P¯rt(1+τy)Y¯t=exp(g¯Y(ρ1)g¯Z)1+R¯h(x¯m)(v¯ηm+u¯exp((ρ1)g¯Zg¯M))(124)(1τe)P¯rt(1+τy)Y¯t=exp(g¯Y(ρ1)g¯Z)1+R¯(v¯ηe+u¯exp((ρ1)g¯Zg¯M))
where v ¯ v ¯ bar(v)\bar{v}v¯ and u ¯ u ¯ bar(u)\bar{u}u¯ satisfy
(125) v ¯ = μ 1 μ ( 1 τ c ) P ¯ r t ( 1 + τ y ) Y ¯ t + exp ( g ¯ Y ( ρ 1 ) g ¯ Z ) 1 + R ¯ v ¯ ( 1 δ ¯ c ) ζ ( x ¯ c ) (126) u ¯ = ( 1 τ m ) P ¯ r t ( 1 + τ y ) Y ¯ t x ¯ m + exp ( g ¯ Y ( ρ 1 ) g ¯ Z ) 1 + R ¯ ( h ( x m t ) η m v ¯ + ( 1 δ ¯ c + h ( x ¯ m ) ) u ¯ exp ( ( ρ 1 ) g ¯ Z g ¯ M ) ) (125) v ¯ = μ 1 μ 1 τ c P ¯ r t 1 + τ y Y ¯ t + exp g ¯ Y ( ρ 1 ) g ¯ Z 1 + R ¯ v ¯ 1 δ ¯ c ζ x ¯ c (126) u ¯ = 1 τ m P ¯ r t 1 + τ y Y ¯ t x ¯ m + exp g ¯ Y ( ρ 1 ) g ¯ Z 1 + R ¯ h x m t η m v ¯ + 1 δ ¯ c + h x ¯ m u ¯ exp ( ρ 1 ) g ¯ Z g ¯ M {:[(125) bar(v)=(mu-1)/(mu)-((1-tau_(c)) bar(P)_(rt))/((1+tau_(y)) bar(Y)_(t))+(exp( bar(g)_(Y)-(rho-1) bar(g)_(Z)))/(1+( bar(R))) bar(v)(1- bar(delta)_(c))zeta( bar(x)_(c))],[(126) bar(u)=-((1-tau_(m)) bar(P)_(rt))/((1+tau_(y)) bar(Y)_(t)) bar(x)_(m)+(exp( bar(g)_(Y)-(rho-1) bar(g)_(Z)))/(1+( bar(R)))(h(x_(mt))eta_(m)( bar(v))+(1- bar(delta)_(c)+h( bar(x)_(m)))( bar(u))exp((rho-1) bar(g)_(Z)- bar(g)_(M)))]:}\begin{gather*} \bar{v}=\frac{\mu-1}{\mu}-\frac{\left(1-\tau_{c}\right) \bar{P}_{r t}}{\left(1+\tau_{y}\right) \bar{Y}_{t}}+\frac{\exp \left(\bar{g}_{Y}-(\rho-1) \bar{g}_{Z}\right)}{1+\bar{R}} \bar{v}\left(1-\bar{\delta}_{c}\right) \zeta\left(\bar{x}_{c}\right) \tag{125}\\ \bar{u}=-\frac{\left(1-\tau_{m}\right) \bar{P}_{r t}}{\left(1+\tau_{y}\right) \bar{Y}_{t}} \bar{x}_{m}+\frac{\exp \left(\bar{g}_{Y}-(\rho-1) \bar{g}_{Z}\right)}{1+\bar{R}}\left(h\left(x_{m t}\right) \eta_{m} \bar{v}+\left(1-\bar{\delta}_{c}+h\left(\bar{x}_{m}\right)\right) \bar{u} \exp \left((\rho-1) \bar{g}_{Z}-\bar{g}_{M}\right)\right) \tag{126} \end{gather*}(125)v¯=μ1μ(1τc)P¯rt(1+τy)Y¯t+exp(g¯Y(ρ1)g¯Z)1+R¯v¯(1δ¯c)ζ(x¯c)(126)u¯=(1τm)P¯rt(1+τy)Y¯tx¯m+exp(g¯Y(ρ1)g¯Z)1+R¯(h(xmt)ηmv¯+(1δ¯c+h(x¯m))u¯exp((ρ1)g¯Zg¯M))
In equilibrium, it must be that v ¯ 0 v ¯ 0 bar(v) >= 0\bar{v} \geq 0v¯0 and u ¯ 0 u ¯ 0 bar(u) >= 0\bar{u} \geq 0u¯0.
Using equations (122), (123), and (124), Θ c , Θ m Θ c , Θ m Theta_(c),Theta_(m)\Theta_{c}, \Theta_{m}Θc,Θm and Θ e Θ e Theta_(e)\Theta_{e}Θe can be expressed as
Θ c = [ 1 τ c ρ 1 ] [ 1 + R ¯ exp ( g ¯ Y ) ] i ¯ r v ¯ Θ m = [ 1 τ m ρ 1 ] [ 1 δ m ζ ( x ¯ c ) η m ] [ 1 + R ¯ exp ( g ¯ Y ) ] i ¯ r v ¯ 1 [ 1 + u ¯ v ¯ η m exp ( ( ρ 1 ) g ¯ Z g ¯ M ) ] Θ e = [ 1 τ e ρ 1 ] [ 1 δ e ζ ( x ¯ c ) η e ] [ 1 + R ¯ exp ( g ¯ Y ) ] i ¯ r v ¯ 1 [ 1 + u ¯ v ¯ η e exp ( ( ρ 1 ) g ¯ Z g ¯ M ) ] Θ c = 1 τ c ρ 1 1 + R ¯ exp g ¯ Y i ¯ r v ¯ Θ m = 1 τ m ρ 1 1 δ m ζ x ¯ c η m 1 + R ¯ exp g ¯ Y i ¯ r v ¯ 1 1 + u ¯ v ¯ η m exp ( ρ 1 ) g ¯ Z g ¯ M Θ e = 1 τ e ρ 1 1 δ e ζ x ¯ c η e 1 + R ¯ exp g ¯ Y i ¯ r v ¯ 1 1 + u ¯ v ¯ η e exp ( ρ 1 ) g ¯ Z g ¯ M {:[Theta_(c)=[(1-tau_(c))/(rho-1)][(1+( bar(R)))/(exp( bar(g)_(Y)))]( bar(i)_(r))/(( bar(v)))],[Theta_(m)=[(1-tau_(m))/(rho-1)][1-delta_(m)(zeta( bar(x)_(c)))/(eta_(m))][(1+( bar(R)))/(exp( bar(g)_(Y)))]( bar(i)_(r))/(( bar(v)))(1)/([1+(( bar(u)))/(( bar(v))eta_(m))exp((rho-1) bar(g)_(Z)- bar(g)_(M))])],[Theta_(e)=[(1-tau_(e))/(rho-1)][1-delta_(e)(zeta( bar(x)_(c)))/(eta_(e))][(1+( bar(R)))/(exp( bar(g)_(Y)))]( bar(i)_(r))/(( bar(v)))(1)/([1+(( bar(u)))/(( bar(v))eta_(e))exp((rho-1) bar(g)_(Z)- bar(g)_(M))])]:}\begin{gathered} \Theta_{c}=\left[\frac{1-\tau_{c}}{\rho-1}\right]\left[\frac{1+\bar{R}}{\exp \left(\bar{g}_{Y}\right)}\right] \frac{\bar{i}_{r}}{\bar{v}} \\ \Theta_{m}=\left[\frac{1-\tau_{m}}{\rho-1}\right]\left[1-\delta_{m} \frac{\zeta\left(\bar{x}_{c}\right)}{\eta_{m}}\right]\left[\frac{1+\bar{R}}{\exp \left(\bar{g}_{Y}\right)}\right] \frac{\bar{i}_{r}}{\bar{v}} \frac{1}{\left[1+\frac{\bar{u}}{\bar{v} \eta_{m}} \exp \left((\rho-1) \bar{g}_{Z}-\bar{g}_{M}\right)\right]} \\ \Theta_{e}=\left[\frac{1-\tau_{e}}{\rho-1}\right]\left[1-\delta_{e} \frac{\zeta\left(\bar{x}_{c}\right)}{\eta_{e}}\right]\left[\frac{1+\bar{R}}{\exp \left(\bar{g}_{Y}\right)}\right] \frac{\bar{i}_{r}}{\bar{v}} \frac{1}{\left[1+\frac{\bar{u}}{\bar{v} \eta_{e}} \exp \left((\rho-1) \bar{g}_{Z}-\bar{g}_{M}\right)\right]} \end{gathered}Θc=[1τcρ1][1+R¯exp(g¯Y)]i¯rv¯Θm=[1τmρ1][1δmζ(x¯c)ηm][1+R¯exp(g¯Y)]i¯rv¯1[1+u¯v¯ηmexp((ρ1)g¯Zg¯M)]Θe=[1τeρ1][1δeζ(x¯c)ηe][1+R¯exp(g¯Y)]i¯rv¯1[1+u¯v¯ηeexp((ρ1)g¯Zg¯M)]
where ζ ( x ¯ c ) η m = a v s i z e a v z ¯ i z e m , ζ ( x ¯ c ) η e = a v s ¯ i z e c a v s ¯ i z e e , η m exp ( g ¯ M ( ρ 1 ) g ¯ Z ) = avsize m ζ x ¯ c η m = a v s i z e ¯ a v z ¯ i z e m , ζ x ¯ c η e = a v s ¯ i z e c a v s ¯ i z e e , η m exp g ¯ M ( ρ 1 ) g ¯ Z = avsize m (zeta( bar(x)_(c)))/(eta_(m))=(av bar(size))/(av( bar(z))ize_(m)),(zeta( bar(x)_(c)))/(eta_(e))=(av( bar(s))ize_(c))/(av( bar(s))ize_(e)),eta_(m)exp( bar(g)_(M)-(rho-1) bar(g)_(Z))=avsize_(m)\frac{\zeta\left(\bar{x}_{c}\right)}{\eta_{m}}=\frac{a v \overline{s i z e}}{a v \bar{z} i z e_{m}}, \frac{\zeta\left(\bar{x}_{c}\right)}{\eta_{e}}=\frac{a v \bar{s} i z e_{c}}{a v \bar{s} i z e_{e}}, \eta_{m} \exp \left(\bar{g}_{M}-(\rho-1) \bar{g}_{Z}\right)=\operatorname{avsize_{m}}ζ(x¯c)ηm=avsizeavz¯izem,ζ(x¯c)ηe=avs¯izecavs¯izee,ηmexp(g¯M(ρ1)g¯Z)=avsizem and η e exp ( g ¯ M η e exp g ¯ M eta_(e)exp( bar(g)_(M)-:}\eta_{e} \exp \left(\bar{g}_{M}-\right.ηeexp(g¯M ( ρ 1 ) g ¯ Z ) = a v s i z e e ( ρ 1 ) g ¯ Z = a v s i z e ¯ e {:(rho-1) bar(g)_(Z))=a_(v) bar(size)_(e)\left.(\rho-1) \bar{g}_{Z}\right)=a_{v} \overline{s i z e}_{e}(ρ1)g¯Z)=avsizee as in our baseline model. These expressions coincide with expressions 86 , ( 87 ) 86 , ( 87 ) 86,(87)86,(87)86,(87) and (88) in our baseline model with the exception of the added terms [ 1 + u ¯ v ¯ × a v s ¯ i z e m ] 1 1 + u ¯ v ¯ × a v s ¯ i z e m 1 [1+(( bar(u)))/(( bar(v))xx av( bar(s))ize_(m))]^(-1)\left[1+\frac{\bar{u}}{\bar{v} \times a v \bar{s} i z e_{m}}\right]^{-1}[1+u¯v¯×avs¯izem]1 and [ 1 + u ¯ v ¯ × a v s ¯ i z e e ] 1 1 + u ¯ v ¯ × a v s ¯ i z e e 1 [1+(( bar(u)))/(( bar(v))xx av( bar(s))ize_(e))]^(-1)\left[1+\frac{\bar{u}}{\bar{v} \times a v \bar{s} i z e_{e}}\right]^{-1}[1+u¯v¯×avs¯izee]1 in Θ m Θ m Theta_(m)\Theta_{m}Θm and Θ e Θ e Theta_(e)\Theta_{e}Θe. The ranking of impact elasticities is equal to that in our baseline specification. Lemma 4 providing conditions under which the equilibrium allocation of innovative investment on the initial BGP is conditionally efficient holds, except that in case (i) there is an additional requirement that u ¯ 0 u ¯ ¯ 0 bar(bar(u))rarr0\overline{\bar{u}} \rightarrow 0u¯0.
We now describe how we infer the value of P ¯ r + x ¯ e Y ¯ t P ¯ r + x ¯ e Y ¯ t ( bar(P)_(r)+ bar(x)_(e))/( bar(Y)_(t))\frac{\bar{P}_{r}+\bar{x}_{e}}{\bar{Y}_{t}}P¯r+x¯eY¯t, which we use to measure Θ e Θ e Theta_(e)\Theta_{e}Θe in equation (36) (and which is also used to measure the remaining impact elasticities). By the free entry condition (124),
(127) P ¯ r t x ¯ e ( 1 + τ y ) Y ¯ t = exp ( g ¯ Y ) 1 + R ¯ v ¯ s ¯ e ( 1 τ e ) ( 1 + u ¯ v ¯ × a v s ¯ i z e e ) (127) P ¯ r t x ¯ e 1 + τ y Y ¯ t = exp ( g ¯ Y ) 1 + R ¯ v ¯ s ¯ e 1 τ e 1 + u ¯ v ¯ × a v s ¯ i z e e {:(127)( bar(P)_(rt) bar(x)_(e))/((1+tau_(y)) bar(Y)_(t))=(exp(( bar(g))Y))/(1+( bar(R)))(( bar(v)) bar(s)_(e))/((1-tau_(e)))(1+(( bar(u)))/(( bar(v))xx av( bar(s))ize_(e))):}\begin{equation*} \frac{\bar{P}_{r t} \bar{x}_{e}}{\left(1+\tau_{y}\right) \bar{Y}_{t}}=\frac{\exp (\bar{g} Y)}{1+\bar{R}} \frac{\bar{v} \bar{s}_{e}}{\left(1-\tau_{e}\right)}\left(1+\frac{\bar{u}}{\bar{v} \times a v \bar{s} i z e_{e}}\right) \tag{127} \end{equation*}(127)P¯rtx¯e(1+τy)Y¯t=exp(g¯Y)1+R¯v¯s¯e(1τe)(1+u¯v¯×avs¯izee)
where s ¯ e s ¯ e bar(s)_(e)\bar{s}_{e}s¯e and a v s ¯ s i z e e a v s ¯ s i z e ¯ e av bar(s) bar(size)_(e)a v \bar{s} \overline{s i z e}_{e}avs¯sizee can be measured as described in Appendix C.3. Combining equations (125) and (126), we obtain
(128) v ¯ ( 1 + u ¯ v ¯ ) = μ 1 μ ( 1 τ c ) P ¯ r t ( x ¯ c + x ¯ m ) ( 1 + τ y ) Y ¯ t + exp ( g ¯ Y ) 1 + R ¯ v ¯ ( 1 s ¯ e ) [ 1 + u ¯ v ¯ × avsize c m ] (128) v ¯ 1 + u ¯ v ¯ = μ 1 μ 1 τ c P ¯ r t x ¯ c + x ¯ m 1 + τ y Y ¯ t + exp ( g ¯ Y ) 1 + R ¯ v ¯ 1 s ¯ e 1 + u ¯ v ¯ × avsize c m {:(128) bar(v)(1+(( bar(u)))/(( bar(v))))=(mu-1)/(mu)-((1-tau_(c)) bar(P)_(rt)( bar(x)_(c)+ bar(x)_(m)))/((1+tau_(y)) bar(Y)_(t))+(exp(( bar(g))Y))/(1+( bar(R))) bar(v)(1- bar(s)_(e))[1+(( bar(u)))/(( bar(v))xxavsize_(cm))]:}\begin{equation*} \bar{v}\left(1+\frac{\bar{u}}{\bar{v}}\right)=\frac{\mu-1}{\mu}-\frac{\left(1-\tau_{c}\right) \bar{P}_{r t}\left(\bar{x}_{c}+\bar{x}_{m}\right)}{\left(1+\tau_{y}\right) \bar{Y}_{t}}+\frac{\exp (\bar{g} Y)}{1+\bar{R}} \bar{v}\left(1-\bar{s}_{e}\right)\left[1+\frac{\bar{u}}{\bar{v} \times \operatorname{avsize}_{c m}}\right] \tag{128} \end{equation*}(128)v¯(1+u¯v¯)=μ1μ(1τc)P¯rt(x¯c+x¯m)(1+τy)Y¯t+exp(g¯Y)1+R¯v¯(1s¯e)[1+u¯v¯×avsizecm]
where
avsize c m = ( 1 s ¯ e ) ( 1 f ¯ e )  avsize  c m = 1 s ¯ e 1 f ¯ e " avsize "_(cm)=((1- bar(s)_(e)))/((1- bar(f)_(e)))\text { avsize }_{c m}=\frac{\left(1-\bar{s}_{e}\right)}{\left(1-\bar{f}_{e}\right)} avsize cm=(1s¯e)(1f¯e)
Given a value of u ¯ / v ¯ u ¯ / v ¯ bar(u)// bar(v)\bar{u} / \bar{v}u¯/v¯ and measures of s ¯ e s ¯ e bar(s)_(e)\bar{s}_{e}s¯e, a v s ¯ i z e c m a v s ¯ i z e c m av bar(s)ize_(cm)a v \bar{s} i z e_{c m}avs¯izecm, and P ¯ r t ( x ¯ c + x ¯ m ) ( 1 + τ y ) P ¯ r t x ¯ c + x ¯ m 1 + τ y ( bar(P)_(rt)( bar(x)_(c)+ bar(x)_(m)))/((1+tau_(y)))\frac{\bar{P}_{r t}\left(\bar{x}_{c}+\bar{x}_{m}\right)}{\left(1+\tau_{y}\right)}P¯rt(x¯c+x¯m)(1+τy), we use equation (127) to calculate v ¯ v ¯ bar(v)\bar{v}v¯ and P ¯ r x ¯ e ( 1 + τ y ) Y ¯ t P ¯ r x ¯ e 1 + τ y Y ¯ t ( bar(P)_(r) bar(x)_(e))/((1+tau_(y)) bar(Y)_(t))\frac{\bar{P}_{r} \bar{x}_{e}}{\left(1+\tau_{y}\right) \bar{Y}_{t}}P¯rx¯e(1+τy)Y¯t, with which we can then calculate all impact elasticities.
While the ratio u ¯ / v ¯ u ¯ / v ¯ bar(u)// bar(v)\bar{u} / \bar{v}u¯/v¯ cannot be inferred directly, we can bound it as follows. Equation (125) provides an upper bound for v ¯ / ( v ¯ + u ¯ ) v ¯ / ( v ¯ + u ¯ ) bar(v)//( bar(v)+ bar(u))\bar{v} /(\bar{v}+\bar{u})v¯/(v¯+u¯) (since x ¯ c 0 x ¯ c 0 bar(x)_(c) >= 0\bar{x}_{c} \geq 0x¯c0 ), and equation (126) provides a lower bound for v ¯ / ( v ¯ + u ¯ ) v ¯ / ( v ¯ + u ¯ ) bar(v)//( bar(v)+ bar(u))\bar{v} /(\bar{v}+\bar{u})v¯/(v¯+u¯) (since x ¯ m 0 ) x ¯ m 0 {: bar(x)_(m) >= 0)\left.\bar{x}_{m} \geq 0\right)x¯m0). Given v ¯ / ( v ¯ + u ¯ ) v ¯ / ( v ¯ + u ¯ ) bar(v)//( bar(v)+ bar(u))\bar{v} /(\bar{v}+\bar{u})v¯/(v¯+u¯), we can pin down the value of x ¯ c x ¯ c bar(x)_(c)\bar{x}_{c}x¯c relative to x ¯ m x ¯ m bar(x)_(m)\bar{x}_{m}x¯m.
Finally, note that using equations (124), (125), and (126), the total value of incumbent firms relative to output in the BGP is given by
v ¯ + u ¯ = 1 ( 1 exp ( g ¯ r ) 1 + R ¯ ) [ μ 1 μ P ¯ r t ( 1 + τ y ) Y ¯ t ( ( 1 τ c ) x ¯ c + ( 1 τ m ) x ¯ m + ( 1 τ e ) x ¯ e ) ] v ¯ + u ¯ = 1 1 exp ( g ¯ r ) 1 + R ¯ μ 1 μ P ¯ r t 1 + τ y Y ¯ t 1 τ c x ¯ c + 1 τ m x ¯ m + 1 τ e x ¯ e bar(v)+ bar(u)=(1)/((1-(exp(( bar(g))r))/(1+( bar(R)))))[(mu-1)/(mu)-( bar(P)_(rt))/((1+tau_(y)) bar(Y)_(t))((1-tau_(c)) bar(x)_(c)+(1-tau_(m)) bar(x)_(m)+(1-tau_(e)) bar(x)_(e))]\bar{v}+\bar{u}=\frac{1}{\left(1-\frac{\exp (\bar{g} r)}{1+\bar{R}}\right)}\left[\frac{\mu-1}{\mu}-\frac{\bar{P}_{r t}}{\left(1+\tau_{y}\right) \bar{Y}_{t}}\left(\left(1-\tau_{c}\right) \bar{x}_{c}+\left(1-\tau_{m}\right) \bar{x}_{m}+\left(1-\tau_{e}\right) \bar{x}_{e}\right)\right]v¯+u¯=1(1exp(g¯r)1+R¯)[μ1μP¯rt(1+τy)Y¯t((1τc)x¯c+(1τm)x¯m+(1τe)x¯e)]
which coincides with the expression (91) for v ¯ v ¯ bar(v)\bar{v}v¯ in our baseline model.

References

Akcigit, Ufuk and William R. Kerr, "Growth through Heterogeneous Innovations," Journal of Political Economy, 2018, 126 (4), 1374-1443.
Garcia-Macia, Daniel, Pete Klenow, and Chang-Tai Hsieh, "How Destructive is Innovation?," Working Paper 22953, National Bureau of Economic Research, December 2016.
Hall, Robert E., "Corporate Earnings Track the Competitive Benchmark," Working Paper 10150, National Bureau of Economic Research, November 2003.
Klette, Tor Jakob and Samuel Kortum, "Innovating Firms and Aggregate Innovation," Journal of Political Economy, October 2004, 112 (5), 986-1018.
Lentz, Rasmus and Dale T. Mortensen, "An Empirical Model of Growth Through Product Innovation," Econometrica, November 2008, 76 (6), 1317-1373.
_ and _ , "Optimal Growth Through Product Innovation," Review of Economic Dynamics, January 2016, 19, 4-19.
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Peters, Michael, "Heterogeneous Markups, Growth and Endogenous Misallocation," Unpublished Paper, Yale University 2016.
Poterba, James M., "The Rate of Return to Corporate Capital and Factor Shares: New Estimates Using Revised National Income Accounts and Capital Stock Data," CarnegieRochester Conference Series on Public Policy, June 1998, 48, 211-246.
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  1. *UCLA and Federal Reserve Bank of Minneapolis
    ^(†){ }^{\dagger} UCLA
  2. 27 27 ^(27){ }^{27}27 The agents' intertemporal elasticity of substitution is used only in computing the transition dynamics of physical capital, for given changes in research labor as we do in our policy counterfactuals. Hence, our results regarding the transition dynamics of aggregate productivity do not depend on this parameter. Moreover, our results on welfare, to a first approximation, do not depend on this parameter. Only the transition dynamics of aggregate output conditional on the transition dynamics of aggregate productivity are affected by the intertemporal elasticity of substitution.
  3. 28 28 ^(28){ }^{28}28 This value of production subsidy is set so that ( 1 τ corp ) ( ( 1 + τ y ) α y μ k ¯ d k ) = α y k ¯ d k = R ¯ 1 τ corp  1 + τ y α y μ k ¯ ¯ d k = α y k ¯ d k = R ¯ (1-tau_("corp "))((1+tau_(y))(alpha y)/(mu) bar(bar(k))-d_(k))=alpha(y)/(( bar(k)))-d_(k)= bar(R)\left(1-\tau_{\text {corp }}\right)\left(\left(1+\tau_{y}\right) \frac{\alpha y}{\mu} \overline{\bar{k}}-d_{k}\right)=\alpha \frac{y}{\bar{k}}-d_{k}=\bar{R}(1τcorp )((1+τy)αyμk¯dk)=αyk¯dk=R¯.
  4. 29 29 ^(29){ }^{29}29 If we set ζ 2 = 0.05 ζ 2 = 0.05 zeta_(2)=0.05\zeta_{2}=0.05ζ2=0.05, close to its lower bound, then Θ = 0.0098 Θ = 0.0098 Theta=0.0098\Theta=0.0098Θ=0.0098.
  5. 30 30 ^(30){ }^{30}30 The Matlab codes that we use to solve the model numerically (linearly and nonlinearly) are available at www.econ.ucla.edu/arielb/innovationpolicycodes.zip.
  6. 32 32 ^(32){ }^{32}32 Klette and Kortum (2004) consider an extension of their model in which firms differ permanently in terms of the size of their step sizes (and markups) on the products they own. With ρ = 1 ρ = 1 rho=1\rho=1ρ=1, markup heterogeneity does not affect aggregate productivity. Since the cost of innovation also scales up with step size (or equivalently, the probability of success falls with step size), all firms choose the same innovation investment per product.
  7. 33 33 ^(33){ }^{33}33 In our baseline model, we have assumed that innovations associated with product z z zzz displace other products with productivity z, whereas here (as well as in Klette and Kortum 2004) we have assumed that displaced products are drawn randomly from the entire distribution (see also Appendix E.4). When ρ = 1 ρ = 1 rho=1\rho=1ρ=1, these two assumptions have identical aggregate implications since size is independent of z z zzz.
  8. 34 34 ^(34){ }^{34}34 To match the quantitative magnitude of these effects, Akcigit and Kerr (2018) argue that investments by incumbent firms to acquire new products scale moderately slower than the number of products in the firm. As a result, the relationship between aggregate productivity growth and aggregate innovative investment in their estimated model is a function of the firm size distribution. Thus, in order to assess the impact of innovation policies on the dynamics of aggregate productivity, one would have to solve their model fully numerically.